Dedicated to celebrate Francisco (Paco) Marcellán´s 60-th birthday
Universidad Carlos III de Madrid
August 29 to September 2, 2011
Sponsors
Universidad Carlos III de Madrid
Society for Industrial and Applied Mathematics
Universidad Politécnica de Madrid
Sociedad Española de Matemáticas Aplicadas
Universidad de Zaragoza
Institute for the Mathematical Sciences
Proyecto Ingenio Mathematica (i-MATH)
Real Sociedad Matemática Española
Ministerio de Ciencia e Innovación
Contents
Contents
1
1
Paco’s biographical notes
3
2
Conference Schedule
5
3
Monday, August 29
3.1 Schedule of the day . . . . .
3.2 Plenary lectures . . . . . . .
3.3 Short communications . . . .
3.3.1 Room A (Auditorium)
3.3.2 Room B (4.1.E01) . .
3.3.3 Room C (4.1.E03) . .
3.3.4 Room D (4.1.E04) . .
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7
7
7
9
9
11
13
15
Tuesday, August 30
4.1 Schedule of the day . . . . .
4.2 Plenary lectures . . . . . . .
4.3 Short communications . . . .
4.3.1 Room A (Auditorium)
4.3.2 Room B (4.1.E01) . .
4.3.3 Room C (4.1.E03) . .
4.3.4 Room D (4.1.E04) . .
4.4 Posters . . . . . . . . . . . .
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19
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24
27
29
32
Wednesday, August 31
5.1 Schedule of the day . . . . .
5.2 Plenary lectures . . . . . . .
5.3 Short communications . . . .
5.3.1 Room A (Auditorium)
5.3.2 Room B (4.1.E01) . .
5.3.3 Room C (4.1.E03) . .
5.3.4 Room D (4.1.E04) . .
5.4 Posters . . . . . . . . . . . .
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37
37
37
38
38
39
40
42
43
4
5
6
Thursday, September 1
47
1
2
CONTENTS
6.1
6.2
6.3
Schedule of the day . . . . .
Plenary lectures . . . . . . .
Short communications . . . .
6.3.1 Room A (Auditorium)
6.3.2 Room B (4.1.E01) . .
6.3.3 Room C (4.1.E03) . .
6.3.4 Room D (4.1.E04) . .
Posters . . . . . . . . . . . .
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47
48
49
49
52
56
59
62
Friday, September 2
7.1 Schedule of the day . . . . .
7.2 Plenary lectures . . . . . . .
7.3 Short communications . . . .
7.3.1 Room A (Auditorium)
7.3.2 Room B (4.1.E01) . .
7.3.3 Room C (4.1.E03) . .
7.3.4 Room D (4.1.E04) . .
7.4 Posters . . . . . . . . . . . .
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67
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8
Committees
8.1 Scientific Committee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Organizing Committee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Local Committee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
77
77
78
9
List of Participants
79
6.4
7
10 Campus Map
93
1
Francisco (Paco) Marcellán Español. Biographical
notes
Francisco (Paco) Marcellán Español was born on September 15, 1951,
in the city of Zaragoza located in northern Spain. Son of Jose Marı́a
Marcellán Alcubierre and Alicia Español Dı́az, he spent his childhood
in the town of Jaca, on the outskirts of the Pyrenees, where his father, with medical training, had been stationed on a military position.
There he studied primary and secondary schools. At the age of 17,
in October 1968, he enrolled in the Faculty of Sciences, at Zaragoza
University, to study Mathematics. Being from another town, he lived
in the students dormitory ”Colegio Mayor Cerbuna” which many of
us have visited.
The five years spent at the Faculty marked his future in many ways.
On one hand, he met and married his lifelong companion Rosa
Fernández Cifuentes, with whom he has two beautiful daughters, Alba
and Clara. On the other, he forged strong ties, which persist to the
present, with fellow students that shared his views on the necessity of
eradicating the totalitarian regime and building up a modern democratic Spain with room for all political tendencies. His activism led
him to be held under custody for questioning on several occasions.
He graduated 5 years later in June 1973 with outstanding marks for which he received Awards from
the University and on a National scale. He initiated immediately his professional career at the University
of Zaragoza combining teaching and research under the supervision of the late Prof. Luis Vigil y Vázquez,
founder of the Spanish School on Orthogonal Polynomials. Paco defended his Ph. D. thesis ”Orthogonal
polynomials on lemniscates” on December 1976, in the middle of a sit-in at the Faculty in support of a
colleague who had been unjustly treated. From Zaragoza, he moved as Full Professor to Universidad de
Santiago de Compostela (1981-82), Universidad Politécnica de Madrid (1982-1991) and, finally, Universidad
Carlos III de Madrid (1991 until present).
Since then, Paco has spread the seed of Orthogonal Polynomials throughout Spain. He has supervised
7 Master’s thesis and 30 Ph. D. thesis, of which 13 were by females and 18 males. Of his students 21
were Spanish, 3 Portuguese, 2 Cubans, and 1 from Belgium, Colombia, Mexico, Venezuela, and Morocco
each. The list of his papers in peer reviewed journals exceeds 190 and is coauthor of several textbooks.
His teaching and scientific activities have been complemented by an intense organizational work. He
held successively the posts of Vice-rector for Research at Universidad Carlos III de Madrid (1995-2004),
Director of the National Agency for Quality Assessment and Accreditation (2004-06), and Secretary General
3
4
CHAPTER 1. PACO’S BIOGRAPHICAL NOTES
for Scientific and Technological Policy (2006-2008). At present he is the Head of the Department of
Mathematics at Universidad Carlos III de Madrid. Between 1991 and 1995 he was the first Head of the
Department of Engineering. Paco has also been actively involved in the SIAM Activity Group on Orthogonal
Polynomials and Special Functions as programme Director (1999-2004) and Chair (2008-2010).
Amazingly, Paco still finds time for daily marathon training. Since 1988 he has participated uninterruptedly in the Madrid Marathon combining this with other competitions of the same nature in Spain and
abroad. His favorite relaxing activity is the movies which he enjoys in the company of friends with whom
he meets on weekends to chat, after the projection, over some tapas and Spanish wine, for which he is a
pretty good expert.
Guillermo López Lagomasino
July 15, 2010
2
Conference Schedule
Monday, 29
Opening
Plenary lecture
Plenary lecture
09:00 to 09:30
09:30 to 10:20
10:30 to 11:20
5
11:20 to 12:00
Coffee break
12:00 to 12:25
12:30 to 12:55
13:00 to 13:25
Award of the GSP
Plenary lecture of
the GSP recipient
13:25 to 15:00
15:00
16:00
to 16:50
17:00
17:30
18:00
18:30
19:00
to 15:50
16:00-16:25
16:30-16:55
to 17:25
to 17:55
to 18:25
to 18:55
to 20:00
Tuesday, 30
Wednesday, 31
Thursday, 1
Friday, 2
Plenary lecture
Plenary lecture
Plenary lecture
Plenary lecture
Plenary lecture
Plenary lecture
Plenary lecture
Plenary lecture
PS-I & coffee break
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
PS-II & coffee break
PS-III & coffee break
PS-IV & coffee break
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
Lunch
Plenary lecture
ST ST ST
ST ST ST
ST ST ST
ST ST ST
ST ST ST
Break
Welcome reception
ST
ST
ST
ST
ST
GSP: Gábor Szegö Prize.
ST: Short Talk
Plenary lecture
Plenary
lecture
ST ST ST ST
ST ST ST ST
ST ST ST ST
ST ST ST ST
ST
ST
ST
ST
ST
ST
ST
ST
ST
Lunch + Open PS
Conference
dinner and
excursion
to Segovia
Open PS: Open Problems Session
Plenary lecture
ST ST ST
ST ST ST
ST ST ST
ST ST ST
ST ST ST
Break
Closing cocktail
ST
ST
ST
ST
ST
PS-No.: Posters Session - Number.
• Conference dinner at the Hotel & Restaurant El Rancho de la Aldegüela, Finca de El Rancho. Plaza del Marqués de Lozoya, 3 Torrecaballeros (Segovia).
• Excursion to Segovia. Segovia is a Spanish city of about 55,000 people in the Castile-Leon province of Spain, about an hour north of Madrid. Designated a
UNESCO World Heritage Site, the old city of Segovia is spectacularly situated atop a long, narrow promontory. It contains a wealth of monuments, including
a cathedral, a magnificent ancient Roman aqueduct, and the beautiful fairy-tale spires of the Alcázar, or castle-palace, that towers over the countryside below.
Scattered about the city are a half a dozen Romanesque churches of great interest, and a church that was once a synagogue. Owing to these famous monuments,
Segovia is a very popular tourist destination, especially as a day-trip from Madrid. It is a must-see city for anyone interested in architecture, history, or religion.
And its monuments aside, Segovia is a scenic place, with mountains as a backdrop to its honey-hued, red-roofed churches and buildings.
3
Monday, August 29
3.1 Schedule of the day
09:00 to 09:30
Chair of plenary session
09:30 to 10:20
10:30 to 11:20
11:20 to 12:00
12:00 to 12:25
12:30 to 13:25
13:25 to 15:00
Chair of plenary session
15:00 to 15:50
Room
Chair of session
16:00-16:25
16:30-16:55
17:00 to 17:25
17:30 to 17:55
18:00 to 18:25
19:00 to 20:00
Opening
Jesús Sánchez Dehesa
Plenary lecture by Manuel Alfaro
Plenary lecture by Walter Van Assche
Coffee break
Announcement of the Gábor Szegö Prize winner by Francisco Marcellán
Plenary lecture by Tom Claeys (Gábor Szego prize)
Lunch
Paul Nevai
Plenary lecture by Vilmos Totik
A (Auditorium)
B (4.1.E01)
C (4.1.E03)
D (4.1.E04)
Nico Temme
Miguel Piñar
Pablo González Vera
Valery Kaliagin
Ramón Orive
Teresa Pérez
Adhemar Bultheel
Judit Minguez
Dmitry Karp
Iván Area
Jorn Schneider
Luis Cotrim
Ester Pérez
Mirta Castro
Hendrik De Bie
Ramón F. Estrada
Erik Koelink
Ana Mendes
J.M.L. Bernard
Ruyman Cruz
Javier Segura Hossain Yakhlef
Antje Vollrath
Imed Lamiri
Reception cocktail at Campus
3.2 Plenary lectures
Manuel Alfaro, Universidad de Zaragoza, Spain.
Title: There’s something about Paco.
Abstract
This talk is one of the two introductory lectures about Paco Marcellán. It consists of some aspects of
Paco’s mathematical life together with some biographical notes, activities, hobbies, and more.
7
8
CHAPTER 3. MONDAY, AUGUST 29
Walter Van Assche, Katholieke Universiteit Leuven, Belgium.
Title: El ingenioso hidalgo Don Paco Marcellán Español.
Abstract
I will make an attempt to describe the scientific contributions of Paco Marcellán. This includes orthogonal
polynomials on Cassinians and lemniscates, modifications of orthogonal polynomials, orthogonal polynomials on the unit circle, semiclassical orthogonal polynomials, Sobolev orthogonal polynomials, linear
combinations of orthogonal polynomials, (quadratic, cubic) polynomial mappings, recurrence relations and
differential equations, asymptotic behavior, zeros and electrostatics, etc. Special attention will be paid to
his pivotal role as a coordinator and public relations officer of orthogonal polynomials in Spain.
Tom Claeys (Gábor Szego prize), Université Catholique de Louvain, Belgium.
Title: Phase transitions in the asymptotics for Toeplitz determinants.
Abstract
I will talk about the asymptotic behavior for determinants of large Toeplitz matrices. Those asymptotics
depend on the smoothness properties of the underlying symbol of the Toeplitz matrices. For smooth symbols
that do not wind around the origin, Szegö’s strong limit theorem gives a description of the asymptotics
for the determinants. For symbols with a finite number of Fisher-Hartwig singularities (combining a jump
discontinuity with a root-type singularity), different asymptotics have been obtained. These results give
rise to two types of phase transitions: one occurs when a Szegö symbol is deformed to a symbol with
one Fisher-Hartwig singularity, another one occurs when a symbol is deformed in such a way that two
Fisher-Hartwig singularities merge. I will describe the asymptotics for the Toeplitz determinants in those
phase transitions in terms of special smooth solutions to the fifth Painlevé equation. I will also discuss two
applications where these critical transitions are of interest, one of them has its origin in number theory, the
other is the two-dimensional Ising model.
The talk will be based on joint work with A. Its and I. Krasovsky.
Vilmos Totik, University of Szeged (Hungary) and University of South Florida, USA.
Title: Fast decreasing polynomials and Christoffel functions.
Abstract
The talk will review some recent results on orthogonal polynomials and Christoffel functions where fast
decreasing polynomials (pin polynomials) played an essential role. We shall also review the construction
and the role of fast decreasing polynomials on the real line and on compact subsets of the complex plane.
Applications towards zero distribution of orthogonal polynomials will be given.
3.3. SHORT COMMUNICATIONS
9
3.3 Short communications
3.3.1 Room A (Auditorium)
R. Orive, Universidad de La Laguna, Spain
Hypergeometric polynomials with varying non-classical parameters.
”Critical” cases
Abstract
During the past decade, several papers papers have been devoted to the asymptotics of rescaled Laguerre
(α
polynomials Ln n (nz) , with varying parameters satisfying that limn→∞ αnn = A ∈ R. The cases where A 6= −1
have been solved by means of classical Potential Theory (A. Martinez Finkelshtein, P. Martinez González and
R. Orive, 2001) or by the powerful Riemann-Hilbert approach (A. B. J. Kuijlaars and K. T.-R. McLaughlin,
2001 and 2004). In this talk, the ”critical” case A = −1 will be considered and the corresponding zero
weak asymptotics will be obtained by a different method. Similar critical situations for Jacobi polynomials
with varying non-standard parameters are considered.
** This is a joint work with: C. Dı́az Mendoza.
D. Karp, Institute of Applied Mathematics, Russia
Log-concavity in parameters, two-sided bounds and representations for
hypergeometric and related functions
Abstract
Inequalities for the generalized hypergeometric function are surprisingly rare in the literature. In the talk
we consider several interrelated classes of such inequalities. First we address the following question: under
what conditions on parameters the generalized hypergeometric function is log-concave as a function of
simultaneous shift of its several parameters? Similar question may be asked about log-convexity but the
answer iis then more accessible due to additivity of log-convexity. We give a complete answer for shifts in
one parameter and partial answer for shifts in two parameters for a generic series in product ratios of rising
factorials and gamma functions. Next we discuss generalized Stieltjes transforms of a non-negative measures
summarizing some of their known and new properties. We introduce the notion of exact Stieltjes order and
find this order for generalized hypergeometric function. Finally, we show how generalized Stieltjes transform
representation of generalized hypergeometric function leads to monotony of their ratios, asymptotically
precise two-sided bounds improving and extending previous results due to Y.Luke and B.C.Carlson and to
log-convexity in parameters for negative values of the argument when additivity cannot be applied. We
also formulate several conjectures, some of analytic and some of combinatorial nature.
** This is a joint work with: S. M. Sitnik and S. Kalmykov.
10
CHAPTER 3. MONDAY, AUGUST 29
Ester Pérez Sinusı́a, University of Zaragoza, Spain
The second and third Appell’s functions for one large variable
Abstract
We consider a Mellin convolution integral representation of the second and third Appell’s function and
apply an asymptotic method designed for this kind of integrals to derive new asymptotic expansions of the
Appell’s functions F2 and F3 for one large variable. For certain values of the parameters, some of these
expansions involve logarithmic terms in the asymptotic variables. The results are illustrated with numerical
experiments.
** This is a joint work with: C. Ferreira, J. L. López.
E. Koelink, Radboud Universiteit, The Netherlands
Spectral properties of differential operators using orthogonal polynomials
Abstract
The classical theorem by Bochner classifies the second order differential operators having polynomial eigenfunctions. Generalising Bochner’s approach we look at differential operators for which there exists a suitable
basis of functions tridiagonalising the differential operator. This gives the opportunity to describe the spectrum of the differential operators involved. We illustrate the approach by several examples, and we discuss
generalisations to other types of operators. A particular well-known example is the Schrödinger with the
Morse potential studied by chemists. Other examples to be discussed are the Jacobi function transform
and the Whittaker transform as well as some q-analogues.
** This is a joint work with: M. Ismail.
Javier Segura, Universidad de Cantabria, Spain
Bounds for solutions of monotonic first order differential systems, with
application to special functions
Abstract
Many special functions are solutions of first order linear systems y0 (x) = a(x)y(x) + d(x)w(x), w0 (x) =
b(x)w(x) + e(x)y(x). We obtain bounds for the ratios y(x)/w(x) and the logarithmic derivatives of y(x)
and w(x) for solutions of monotonic systems satisfying certain initial or boundary conditions. We focus
on systems satisfied by functions y(x) = yn (x) and w(x) = yn−1 (x), with yk (x) functions depending on
a parameter k which are also solutions of three-term recurrence relations and second order differential
equations. It is shown how the bounds obtained are related both with the Liouville-Green approximation
for second order ODEs as well as with the asymptotic behavior of the solutions of the recurrence as
n → +∞. For the case d(x)e(x) > 0 sequences of upper and lower bounds can be built by iterating the
recurrence relation; for minimal solutions of the recurrence these are convergent sequences. Turán-type
3.3. SHORT COMMUNICATIONS
11
inequalities can be also established using these bounds. Many special functions are amenable to this type
of analysis, and we give several examples of application: modified Bessel functions, parabolic cylinder
functions (including Hermite polynomials of imaginary variable), Legendre functions of imaginary variable
and Laguerre functions.
3.3.2 Room B (4.1.E01)
T. E. Pérez, Dpto. Matemática Aplicada. Universidad de Granada, Spain
Sobolev–type orthogonal polynomials in several variables
Abstract
Multivariate orthogonal polynomials associated with a Sobolev–type inner product, that is, an inner product
defined by adding the evaluation of the gradients in several points to a measure are studied. Orthogonal
polynomials and kernel functions associated with this new inner product can be explicitly expressed in terms
of those corresponding to the original measure. We apply our results to the Sobolev–type modification of
the two variable classical measure on the unit disk by adding a finite set of gradients evaluated at equally
spaced points on the border. Then, asymptotics of Christoffel functions are studied.
1. C. F. Bracciali, A. M. Delgado, L. Fernández, T. E. Pérez, M. A. Piñar, New steps on Sobolev
orthogonality in two variables, J. Comput. Appl. Math., 235 (2010), pp. 916–926.
2. Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory, 152
(2008), pp. 52–65.
** This is a joint work with: A. M. Delgado and M. A. Piñar.
I. Area, Universidad de Vigo, Spain
Bivariate second–order linear partial difference equations and monic
orthogonal polynomial solutions
Abstract
In this talk we shall present a classification of bivariate second–order linear partial difference equations,
which are admissible, potentially self–adjoint, and of hypergeometric type. Explicit expressions for the
coefficients of the three–term recurrence relations satisfied by monic orthogonal polynomial solutions are
obtained using vector matrix notation. Finally, as illustration, explicit examples related with bivariate
Charlier, Kravchuk, Meixner and Hahn orthogonal polynomials are given.
** This is a joint work with: E. Godoy and J. Rodal.
12
CHAPTER 3. MONDAY, AUGUST 29
M. Castro Smirnova, University of Seville, Spain
Orthogonal matrix polynomials satisfying differential equations with
recurrence coefficients having non-scalar limits
Abstract
We introduce a family of weight matrices W of the form T (t)T ∗ (t), T (t) = eA t eDt , where A is certain
nilpotent matrix and D is a diagonal matrix with negative real entries. The weight matrices W have
arbitrary size N × N and depend on N parameters.
The orthogonal polynomials with respect to this family of weight matrices satisfy a second order
differential equation with differential coefficients that are matrix polynomials F2 , F1 and F0 (independent
of n) of degrees not bigger than 2, 1 and 0 respectively.
For size 2 × 2, we find an explicit expression for a sequence of orthonormal polynomials with respect
to W . In particular, we show that one of the recurrence coefficients for this sequence of orthonormal
polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the
examples studied up to now in the literature.
** This is a joint work with: J. Borrego and A. Durán.
2
Ana Mendes, Instituto Politécnico de Leiria, Portugal
Matrix Sylvester differential equations in the theory of matrix orthogonal
polynomials
Abstract
In this work we find sequences of matrix polynomials whose orthogonality measure, W , satisfy a matrix
Sylvester differential equation,
W 0 A +W B +CW = Pol(z).
This equation arises from the study of sequences of vector orthogonal polynomials whose derivative sequence
is also orthogonal. A sequence of vector polynomials {Bm }m∈N is said to be left-orthogonal with respect to
the vector of linear functions U = [u1 . . . uN ]T if deg Bm = m and (hk U ) (Bm ) = ∆m δn,m , for k = 0, 1, ..., m−1
and n, m ∈ N, with ∆m non-singular and where δn,m is the Kronecker delta, Can be show that {Bm }m∈N is
left-orthogonal with respect to U if and only if {Vm }m∈N such that Bm = Vm (h(x))P0 (x) is left-orthogonal
with respect to F in the matrix
sense, where F is generalized Markov matrix function associated with U
∞ ((h(x))n Ux )(P0 )
defined by F (z) = ∑n=0
with z such that |h(x)| < |z| for every x ∈ L where L = ∪ j=1,...,N supp uxj
zn+1
T
and P0 (x) = 1 x · · · xN−1 .
** This is a joint work with: A. Branquinho.
3.3. SHORT COMMUNICATIONS
13
H. O. Yakhlef, Universidad Carlos III de Madrid, Spain
Asymptotic behaviour of orthogonal matrix polynomials
Abstract
Let α be a square matrix of meaures, and (Pn (x; α))n the associated sequence of orthonormal matrix polynomials satisfying the recurrence relation x Pn (x; α) = An+1 (α)Pn+1 (x; α)+Bn (α)Pn (x; α)+A∗n (α)Pn−1 (x; α), n ≥
def
0. Let dβ (u) = dα(u) + Mδ (u − c), where M is a constant positive definite matrix, δ is the Dirac measure
and c lies outside of the support of dα . We study the asymptotic behavior of the ratio Pn (x; β )Pn−1 (x; α)
under quite general assumption on (An (α))n and (Bn (α))n .
3.3.3 Room C (4.1.E03)
A. Bultheel, K.U.Leuven, Dept. Computer Science, Belgium
Quasi- and pseudo orthogonal rational functions for quadratures on the
positive real line
Abstract
We consider a positive measure on [0, ∞) and a sequence of nested spaces L0 ⊂ L1 ⊂ L2 · · · of rational
functions with prescribed poles in [−∞, 0]. Let ϕk ∈ Lk be the associated sequence of orthogonal rational
functions. The zeros of ϕn can be used as the nodes of a rational Gauss quadrature formula that is
exact for all functions in Ln · Ln−1 , a space of dimension 2n. Quasi- and pseudo-orthogonal functions are
functions in Ln that are orthogonal to some subspace of Ln−1 . Both of them are generated from ϕn and
ϕn−1 and depend on a real parameter τ. Their zeros can be used as the nodes of a rational Gauss-Radau
quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace
of Ln · Ln−1 where the quadrature is exact. The parameter τ is used to fix a node at a pre-assigned point.
The space where the quadratures are exact have dimension 2n − 1 in both cases but it is in Ln−1 · Ln−1 in
the quasi-orthogonal case and it is in Ln · Ln−2 in the pseudo-orthogonal case.
** This is a joint work with: P. González-Vera, E. Hendriksen and O. Njåstad.
J. Schnieder, University of Lübeck, Germany
Schauder bases of optimal degree with Jacobi polynomials
Abstract
In our lecture we show the existence of a polynomial Schauder bases of optimal degree with Jacobi orthogonality. We proof
14
CHAPTER 3. MONDAY, AUGUST 29
Theorem 1. For all ε > 0 and all α, β ≥ − 12 with max{α, β } > − 12 there exists a sequence of polynomials
(pα,β ,n )n∈N0 , such that
Z 1
1.
−1
pα,β , j (x)pα,β ,i (x)(1 − x)α (1 + x)β dx = δi j ,
2.
degpα,β ,n ≤ n(1 + ε) for all n ∈ N0 ,
3.
∀ f ∈ C[−1, 1] : k f − Sn f k∞ → 0
2 max{α,β }+1
1
kSn kC→C ≤ c
,
ε
4.
where we set Sn f (x) := ∑nj=0 c j ( f )pα,β , j (x) with c j ( f ) :=
R1
−1
(n → ∞),
pα,β , j (t) f (t)ωα,β (t)dt.
One central step in proofing the theorem is to establish localisation properties of certain generalised
Kernelfunctions. To do so, we make heavy use of integral representation formulas for Jacobi polynomials, hypergeometric functions and certain recently proved asymptotics for Jacobi polynomials and Bessel
functions, respectively.
H. De Bie, Ghent University, Belgium
Fourier transforms in Clifford analysis
Abstract
I give an overview of recent work on the study of generalized Fourier transforms in Clifford analysis. These
new integral transforms are constructed in such a way that they intertwine the Dirac operator (or one of
its generalizations, the Dunkl Dirac operator resp. the radially deformed Dirac operator) in a suitable way.
The main question in this topic is to find explicit formulas for the kernels of these generalized integral
transforms (as opposed to series representations). I will show how this can be done for the special class of
Clifford-Fourier transforms, where the end result is a family of kernels expressible as finite sums of suitable
Bessel functions. I will also discuss the spectrum and eigenfunctions of the resulting transforms.
J. M. L. Bernard, CEA-DIF, France
Recent progresses on the expansion of incomplete Bessel function in
complex plane
Abstract
The scattering of a dipole radiation by a multimode plane, that can be passive or active, was recently
reduced to the calculus of an incomplete cylindrical Bessel function in complex plane by the author in [1],
in acoustics and in electromagnetism.
The expansions of this special function given by Agrest and Maksimov [2] are not sufficient for an
efficient determination in all sectors of the complex plane, and new results were given in [1].
Recent discussions with Nico Temme [3] concerning the incomplete Bessel function and the leaky aquifer
function, have led the author to develop a novel convergent series and asymptotics [4] that are presented.
References :
3.3. SHORT COMMUNICATIONS
15
[1] J.M.L. Bernard, ’On the expression of the field scattered by a multimode plane’, The Quarterly
Journal of Mechanics and Applied Mathematics, pp. 237-266, 63, 3, 2010.
[2] M.M. Agrest, M.S. Maksimov, ’Theory of incomplete cylindrical functions and their applications’,
springer, New York, 1971.
[3] N. M. Temme, ’The leaky aquifer function revisited’, Int. J. Quantum Chem., 109, pp. 2826-2830,
2009.
[4] J.M.L. Bernard, ’On a novel expression of the field scattered by an arbitrary constant impedance
plane’, submitted to publication.
A. Vollrath, TU Braunschweig, AG Partielle Differentialgleichungen, Deutschland
A new algorithm for fast Fourier Transforms on the rotation group
Abstract
We will discuss an approximate fast algorithm to calculate the discrete Fourier transform on the rotation
group SO(3). Our method needs O(L3 log L + Q) arithmetic operations for a degree-L transform at Q nodes
free of choice, with the desired accuracy.
With the decomposition of SO(3) basis functions, arises a set of orthogonal polynomials closely related to
Jacobi polynomials - the Wigner-d functions. The talk will focus on a new efficient method to calculate
a particular linear transformation that allows us to replace Wigner-d functions of arbitrary orders with
those of low orders and eventually with Chebyshev polynomials. Based on the differential equations, whose
solutions are the Wigner-d functions, we show that the linear mapping for these conversions appears as
the eigenvector matrix of certain semiseparable matrices. This enables us to employ a known divide-andconquer algorithm for symmetric semiseparable eigenproblems together with the fast multipole method to
calculate the desired change of basis.
** This is a joint work with: J. Keiner.
3.3.4 Room D (4.1.E04)
J. Mı́nguez Ceniceros, Universidad de La Rioja, Spain
Mixed type Angelesco - AT systems are perfect
Abstract
On the basis of Angelesco and AT systems we introduce several types of mixed type multiple orthogonal
systems of polynomials. We discuss the normality and the location of zeros of such multiple orthogonal
polynomials.
** This is a joint work with: U. Fidalgo Prieto and S. Medina Peralta.
16
CHAPTER 3. MONDAY, AUGUST 29
L. Cotrim, Escola Superior de Tecnologia e Gestão do IPL, Portugal
Matrix Interpretation of Multiple Orthogonality
Abstract
In this talk we present an algebraic theory of multiple orthogonal polynomials, that enables us to obtain
a three term recurrence relations with matrix coefficients, satisfed by the sequences of vector multiple
orthogonal polynomials. We give some characterizations of multiple orthogonal polynomials including
recurrence relations, a Favard type theorem and a Christoffel-Darboux type formulas. A reinterpretation of
Hermite-Padé approximation in terms of matrix functions is presented.
Finally, we give a Markov’s type theorem for the type I multiple orthogonal polynomials.
** This is a joint work with: A. Branquinho and A. Foulquié Moreno.
R. F. Álvarez-Estrada, Facultad Ciencias Fisicas, Universidad Complutense, Spain
Quantum quartic anharmonic oscillator at equilibrium: generalized Hermite
polynomials
Abstract
We treat a quantum anharmonic oscillator, subject to a potential (V = V (x) = 2−1 mω 2 x2 + (4!)−1 gx4 ), in
the presence of a heat bath at thermal equilibrium at temperature T and with vanishing friction. The
nonequilibrium evolutions of the oscillator are described by the time(t)-dependent distribution functions
W = W (x, p;t), solving the quantum Wigner linear partial differential equation (E. P. Wigner, Phys. Rev.
”On the quantum correction for thermodynamic equilibrium“ Phys. Rev. 40, (1932), 749-759 ): ∂W /∂t =
−(p/m)(∂W /∂ x) + (dV /dx)(∂W /∂ p) − (h̄2 /(3!22 ))(d 3V /dx3 )(∂ 3W /∂ p3 ), which is exact for the potential
V . x and p are the position and momentum variables for the oscillator. m, ω and g are positive constants.
h̄ is Planck’s constant. We construct the t-independent solution Weq = Weq (x, p) of Wigner’s equation,
representing thermal equilibrium of the oscillator with the heat bath, through new series expansions in
terms of standard Hermite polynomials, which differ from and generalize the one obtained ( for high T
and small h̄) by Wigner, and compare the different expansions. We construct recurrently the new family
of orthogonal polynomials HQ,n , orthogonalized in p (for fixed x) by means of Weq as weight function. The
HQ,n ’s generalize the standard Hermite polynomials, used for classical Brownian motion, and differ from
other generalizations previously reported (R. F. Álvarez-Estrada, “Brownian motion, quantum corrections
and a generalization of the Hermite polynomials”. J. of Comput. and Appl. Math. 233, 1453-1461
(2010)). Mathematical properties of Weq and of the HQ,n ’s are reported. The possible interest of the
HQ,n ’s for nonequilibrium evolutions is discussed briefly. (Submitted to OPSFA-11, dedicated to celebrate
F. Marcellan’s 60th birthday).
3.3. SHORT COMMUNICATIONS
17
R. Cruz-Barroso, La Laguna University, Spain
Multiple para-orthogonal polynomials and multiple Szegő quadrature
formulas
Abstract
Multiple type I and type II Orthogonal Polynomials on the Unit Circle (MOPUCs) were introduced and
studied in (1). Two properties are of particular interest:
1. The obtention of a recurrence relation from the solution of a Riemann-Hilbert problem for which the
solution is in terms of MOPUCs.
2. The behavior of the zeros of MOPUCs, that contrary to the single case, may lie outside the unit disk
of the complex plane.
In this talk we start by analyzing the recurrence relation for type I MOPUCs obtained in that paper,
which can be expressed as a generalization of the well known Szegő recurrence relation, given in terms
of the multiple version of the Verblunsky coefficients. From here we introduce and analyze multiple paraorthogonal polynomials and study the possibility of using their zeros as quadrature nodes, which would
provide a multiple version of the well-known Szegő rules. Some examples are given and some open questions
are proposed.
The ideas of this talk are a generalization to the multiple case of very well known results in the theory
of Orthogonal Polynomials on the Unit Circle where Paco has several of the most important and decisive
contributions.
The talk is part of a joint work in progress with Carlos Dı́az-Mendoza and Ramón Orive, from Department of Mathematical Analysis in La Laguna University, Tenerife.
(1) J. Mı́nguez Ceniceros and W. Van Assche, Multiple Orthogonal Polynomials on the Unit Circle.
Constr. Approx. 28 (2008) 173-197.
I. Lamiri, Ecole Supérieure des Sciences et Technologie H. Sousse, Turnisia
d-orthogonality of discrete q-Hermite type polynomials
Abstract
Let P be the vector space of polynomials with coefficients in C and let P 0 be its algebraic dual. We
denote by hu, f i the effect of the functional u ∈ P 0 on the polynomial f ∈ P. A polynomial sequence
{Pn }n≥0 is called a polynomial set (PS, for shorter) if and only if deg Pn = n for all non-negative integer n.
Let d be a positive integer. A PS {Pn }n≥0 is called d-orthogonal (d-OPS, for shorter) with respect to the
d-dimensional vector of functionals Γ = t (Γ0 , Γ1 , . . . , Γd−1 ) if it fulfills:
(
hΓk , Pr Pn i = 0,
r > nd + k, n ∈ N = {0, 1, 2, ...},
hΓk , Pn Pnd+k i 6= 0, n ∈ N,
18
CHAPTER 3. MONDAY, AUGUST 29
for each integer k belonging to {0, 1, . . . , d − 1}. For d = 1, we recognize the well-known notion of orthogonality.
In this talk, we consider the following two PSs:
!
n(d+1)−1
−n , q−n+1 , . . . , q−n+d q
q
(1)
qd+1 ;
,
Pn (x|q) = xn d+1 Φ0
−
xd+1
!
q−n , q−n+1 , . . . , q−n+d d+1 −q(d+1)
(2)
n
Pn (x|q) = x d+1 Φd
,
q ; xd+1
0, . . . , 0
where
r φs
a1 , . . . , ar
b1 , . . . , bs
q; z :=
∞
(a1 , . . . , ar ; q)n
∑ (b1 , . . . , bs ; q)n
n
(−1)n(1+s−r) q(2)(1+s−r)
n=0
j
with (a1 , . . . , ar ; q)n := ∏rj=1 (a j ; q)n , (a; q)0 := 1, (a; q)n := ∏n−1
j=0 (1 − aq ), n
(1)
(2)
We show that the two PSs {Pn }n≥0 and {Pn }n≥0 are d-orthogonal.
zn
,
(q; q)n
> 1.
For d = 1, the considered polynomials are reduced to the known discrete q-Hermite polynomials I and II.
(1)
(2)
For the limiting case q = 1, the PSs {Pn }n≥0 and {Pn }n≥0 tends to the Gould-Hopper polynomials.
For these two PSs, we state many properties: generating function, inversion formula, d-dimensional functional vectors, difference formulas, difference equation, recurrence relation.
4
Tuesday, August 30
4.1 Schedule of the day
Chair of plenary session
09:30 to 10:20
10:30 to 11:20
11:20 to 12:00
Room
Chair of session
12:00 to 12:25
12:30 to 12:55
13:00 to 13:25
13:25 to 15:00
Chair of plenary session
15:00 to 15:50
16:00 to 16:50
Room
Chair of session
17:00 to 17:25
17:30 to 17:55
18:00 to 18:25
18:30 to 18:55
(∗)
1.2.3.4.5.-
Jeff Geronimo
Plenary lecture by Plamen Iliev
Plenary lecture by Doron Lubinski
A (Auditorium)
Olav Njastad
B. Beckermann
Abey López
Valery Kaliagin
A (Auditorium)
Peter Clarkson
Klaas Deschout
Dries Geudens
Arno Kuijlaars
Adrien Hardy
Poster session I(∗) and coffee break
B (4.1.E01)
C (4.1.E03)
Fitouhi Ahmed
Erik Koelink
Lazhar Dhaouadi
Pablo Sánchez
Kamel Brahim
Gwang Hui Kim
Rezan Sevinik
Choonkil Park
Lunch
Arno Kuijlaars
Plenary lecture by Ulises Fidalgo
Plenary lecture by Roderick Wong
B (4.1.E01)
C (4.1.E03)
Mohamed Atia
Luis Velázquez
Tom Koornwinder
Elena Berdysheva
Fokko van de Bult
James Griffin
M. Foupouagnigni
Oscar Ciaurri
Luc Vinet
Raimundas Vidunas
D (4.1.E04)
Henrik L. Pedersen
Brian Simanek
Jacob Christiansen
Milivoje Lukic
D (4.1.E04)
Jose Petronilho
Ridha Sfaxi
Shuhei Kamioka
M. M. Rodrigues
Yves Grandati
The following posters will be presented during the poster session of the day.
Marı́a Pilar Alfaro.
6.- Giovanni Cassatella.
Jung Rye Lee.
7.- Kenier Castillo.
David Batista Elbano.
8.- Alicia Cachafeiro.
Manuel Bello-Hernández. 9.- Jorge A. Borrego Morell
Junot Cacoq.
19
20
CHAPTER 4. TUESDAY, AUGUST 30
4.2 Plenary lectures
Plamen Iliev, Georgia Institute of Technology, USA.
Title: Bispectral commuting partial difference operators for multivariate hypergeometric polynomials.
Abstract
The classical discrete orthogonal polynomials can be characterized by the fact that they are eigenfunctions
of a second-order difference operator. For some families, this operator can be connected to the three-term
recurrence relation via a duality between the variable and the degree index of the polynomials.
One of the main obstacles for the extension of the above theory to higher dimensions is the fact that
the polynomials are no longer uniquely determined by the orthogonality measure (up to a multiplicative
constant). Thus, the recurrence relations, the spectral properties and the duality between the variables and
the degree indices depend on the construction (the specific basis) of orthogonal polynomials. I will discuss
recent results related to the construction and the classification of multivariate orthogonal polynomials which
are eigenfunctions of two bispectral commutative algebras of partial difference operators: one acting on
the variables of the polynomials and the other one on their degree indices.
Doron Lubinsky, Georgia Institute of Technology, USA.
Title: Beta Ensembles via Gauss Quadrature.
Abstract
In the theory of random matrices, one considers probability distributions on the eigenvalues x1 , x2 , ..., xn of
an n × n Hermitian matrix. For β -ensembles, the probability distribution has the form
P(n) (x1 , x2 , ..., xn ) =
1
|V (x1 , x2 , ..., xn )|β dµ (x1 ) dµ (x2 ) ...dµ (xn ) ,
C
where V is the Vandermonde determinant,
V (x1 , x2 , ..., xn ) =
∏
(xk − x j ) .
1≤ j<k≤n
Furthermore, µ is a positive measure on the real line with all finite power moments, while C is a positive
normalization constant, chosen to ensure that P(n) is a probability measure. The positive parameter β (or
sometimes 1/β ) is called the temperature.
The best understood case is the unitary case β = 2, where there are close connections to orthogonal
polynomials. The orthogonal case β = 1 and symplectic case β = 4 are more difficult, but also well
understood.
We present explicit compact expressions for the m point correlation function
R (x1 , x2 , .., xm ) =
1
n!
(n − m)! C
Z
Z
...
|V (x1 , x2, ..., xm ,tm+1 , ...,tn )|β dµn (tm+1 ) ...dµn (tn ) ,
when µn is the measure formed from the n point Gauss quadrature for µ. In the case m = 1, this gives the
β analogue of the Christoffel function of orthogonal polynomials, also known as the density of states. In
the special case β = 2, this yields a (new?) proof of well known determinantal identities for the correlation
functions.
4.3. SHORT COMMUNICATIONS
21
We use these to establish universality limits for discrete β ensembles for any β > 1. We also review the
state of the art for β = 2 and fixed measures µ with compact support.
Ulises Fidalgo Prieto, Universidade de Aveiro (Portugal) and Universidad Carlos III de Madrid, Spain.
Title: A version of the fundamental theorem of algebra for Markov functions and a generalization of
Markov’s theorem.
Abstract
We consider the wide class of Nikishin systems of functions. Such systems are made up of Cauchy transforms of measures (with a special structure) supported on the same interval. We show that generalized
polynomials made up of polynomial combinations of the functions in the Nikishin system verify a version
of the fundamental theorem of algebra. This result has multiple applications in the asymptotic theory of
multiple orthogonal polynomials and in the convergence theory of Hermite-Padé approximation. In particular, we give very general conditions under which the sequence of type II Hermite-Padé approximations
of a Nikishin system of functions converges uniformly on each compact subset of the complement of the
interval supporting the measures.
Roderick Wong, City University of Hong Kong, China.
Title: Asymptotics of Discrete Orthogonal Polynomials.
Abstract
This is a survey of results in asymptotic approximations of discrete orthogonal polynomials. These polynomials include Charlier, Meixner, Krawtchouk, discrete Chebyshev, Hahn and Tricomi-Carlitz. Brief descriptions
on the methods used to derive these approximations will also be given.
4.3 Short communications
4.3.1 Room A (Auditorium)
B. Beckermann, Labo Painlevé, Univ. Lille 1, France
Equilibrium problems for vector potentials with semidefinite interaction
matrices and constrained masses
Abstract
In this talk we prove existence and uniqueness of a solution to the problem of minimizing the logarithmic
energy of vector potentials associated to a d-tuple of positive measures supported on closed subsets of the
complex plane. The assumptions we make on the interaction matrix are weaker than the usual ones and
we also let the masses of the measures vary in a compact subset of Rd+ . We characterize the solution in
terms of variational equations. Finally, we review a few examples taken from the recent literature [1,2,3,4]
that are related to our results.
22
CHAPTER 4. TUESDAY, AUGUST 30
[1] A.I. Aptekarev, V.G. Lysov, Systems of Markov functions generated by graphs and the asymptotics
of their Hermite Padé approximants, Sbornik Mathematics 201:2 (2010) 183-234.
[2] A.I. Aptekarev, A.B.J. Kuijlaars, W. Van Assche, Asymptotics of Hermite-Padé rational approximants for two analytic functions with separated pairs of branch points, Internat. Math. Research
Papers 2008.
[3] F. Balogh, M. Bertola, Regularity of a vector potential problem and its spectral curve, J. Approx.
Theory 161, 353?370 (2009)
[4] S. Delvaux, A.B.J. Kuijlaars, A graph-based equilibrium problem for the limiting distribution of
non-intersecting Brownian motions, Constructive Approximation 32 (2010) 467-512 .
** This is a joint work with: V. Kalyagin, A. C. Matos and F. Wielonsky.
Abey López, Katholieke Universiteit Leuven, Belgium
Generalized three-term recurrences and Nikishin systems on star-like sets
Abstract
We consider sequences of monic polynomials satisfying generalized three-term recurrences of the form
zQn (z) = Qn+1 + an−p Qn−p . Here p is an arbitrary positive integer. We will discuss strong asymptotics
of Qn assuming that the recurrence coefficients have an asymptotically periodic structure. Under certain
conditions, we show that these polynomials are multiple orthogonal with respect to a Nikishin system of
measures constructed on compact star-like sets in the complex plane. An important tool is the study
of generalized eigenvalues of the associated Hessenberg matrix. This work is based on earlier papers
by Aptekarev-Kalyagin-Van Iseghem (The genetic sum’s representation for the moments of a system of
Stieltjes functions and its applications, Constr. Approx. 16 (2000), 487-524), Aptekarev-Kalyagin-Saff
(Higher order three-term recurrences and asymptotics of multiple orthogonal polynomials, Constr. Approx.
30 (2009), 175-223), and Delvaux (Equilibrium problem for the eigenvalues of banded block Toeplitz
matrices, arXiv:1101.2644).
** This is a joint work with: S. Delvaux.
V. Kalyagin, Higher School of Economics, Russia
On approximation of measures for Nikishin systems
Abstract
Simultaneous rational approximations (Hermite-Padé approximations) and multiple orthogonal polynomials
attract increasing attention last decades. One fundamental question on multiple orthogonal polynomials is
the connection between measures of orthogonality and associated recurrence coefficients. This question is
well investigated for usual (not multiple) orthogonal polynomials and is far to be clear for multiple orthogonal
polynomials. In this paper we study the so called inverse spectral problem for the case of Nikishin systems:
4.3. SHORT COMMUNICATIONS
23
recover the measures of orthogonality from the known recurrence coefficients. As a main result we propose
an approximation of the weight functions based on a specific solution of recurrence equation and prove the
convergence of approximations. Our construction is closed to the well known Turan determinants for usual
orthogonal polynomials [1]. The main difficulty of the problem is that in our case we need to recover the
set of measures and not only one measure as in the usual case. Another difficulties lies in the nature of
Nikishin systems, the weight functions are generated by a set of measures with the common support on a
different interval. The intriguing question is to find an approximation to all these measures. This work is
partly supported by RFFI grant 10-01-00682. 1. Geronimo and W. Van Assche Approximating the weight
function for orthogonal polynomials on several intervals, JAT v.65 (1991), pp.341-371
** This is a joint work with: M. Sokolov.
K. Deschout, Katholieke Universiteit Leuven, Belgium
Critical behavior for modified Jacobi-Angelesco polynomials
Abstract
We consider multiple orthogonal polynomials with respect to two modified Jacobi weights on touching
intervals [a, 0] and [0, 1], with a n of the polynomial tend to infinity while the parameter a tends to −1
at a rate of O(n−1/2 ). The polynomials then converge to a certain entire solution of a third order linear
differential equation.
Also we consider the associated determinantal point process, and discuss a new family of limiting kernels
that appears under the double scaling.
D. Geudens, Katholieke Universiteit Leuven, Belgium
New critical behavior in the Hermitian two-matrix model
Abstract
We study the Hermitian two-matrix model
1
exp (−nTr (V (M1 ) +W (M2 ) − τM1 M2 )) dM1 dM2 ,
Zn
defined on couples of n × n matrices, with an Gaussian potential V (x) = x2 /2 and a even quartic potential
W (y) = y4 /4 + ty2 /2. Recently, new multi-critical behavior was discovered for t = −1 and τ = 1. For these
critical parameters the limiting eigenvalue density of M1 vanishes as a square root in the interior of the
support. The talk will be about the nature of this critical point. We show that under an appropriate
double scaling limit the correlation kernel can be expressed in terms of the solution to a 4 × 4 RiemannHilbert problem. This Riemann-Hilbert problem is a generalization of the Riemann-Hilbert problem that
was introduced in [1] in the context of non-intersecting Brownian motions at a tacnode. There is also a
connection with the Hastings-McLeod solution to the Painlevé differential equation. Our approach is based
on multiple orthogonal polynomials and the Deift/Zhou steepest decent method applied to a 4 × 4 matrix
valued Riemann-Hilbert problem.
24
CHAPTER 4. TUESDAY, AUGUST 30
[1] S. Delvaux, A.B.J. Kuijlaars and L. Zhang, Critical behavior of non-intersecting Brownian motions
at a tacnode, Comm. Pure Appl. Math. (2011), Accepted.
** This is a joint work with: M. Duits.
A. Kuijlaars, Katholieke Universiteit Leuven, Belgium
Orthogonal polynomials in the normal matrix model
Abstract
I will discuss a new kind of orthogonality for polynomials that replaces the orthogonality with respect to
certain weights on the complex plane that appears in the normal matrix model. The orthogonal polynomials
have their zeros on curves in the complex planes, which I will describe in the simplest case of a cubic
potential.
** This is a joint work with: P. Bleher (IUPUI Indianapolis).
A. Hardy, Katholieke Universiteit Leuven, Belgium
Vector Equilibrium Problems arising from Random Matrix models
Abstract
Some random matrix models are known to have a limiting eigenvalue distribution, when the size of the
matrix goes to infinity, characterized by an equilibrium problem arising from potential theory. This is the
case for the so-called orthogonal polynomial ensembles, for which eigenvalue distributions are controlled by
kernels involving orthogonal polynomials. Recently, it has been understood that some more complicated
random matrix models (as perturbated models, or multi-matrix models) have their limiting eigenvalue
distributions characterized by vector equilibrium problems, that is equilibrium problems involving more
than one measure. We will discuss some examples, their relation to multiple orthogonal polynomials,
and the technical difficulties in the manipulation of such vector equilibrium problems (as the existence of
minimizers or regularity properties).
** This is a joint work with: A. Kuijlaars.
4.3.2 Room B (4.1.E01)
L. Dhaouadi, IPEI Bizerte, Tunisia
An entropy inequality for the q-Bessel Hahn Fourier transform
Abstract
In this paper we use an I.I. Hirschman entropy argument to give an uncertainty inequality related to the
q-Bessel Hahn Fourier transform.
4.3. SHORT COMMUNICATIONS
25
K. Brahim, Faculty of sciences Tunis, Tunisia
Q-Gamma functions
Abstract
We study some properties and inequalities for the q-gamma and the q-beta. We also introduce a symmetric
q-gamma function.
** This is a joint work with: F. Ahmed.
R. Sevinik Adgüzel, Department of Mathematics METU, Turkey
On the q-analysis of the q-hypergeometric difference equation
Abstract
In this study, a fairly detailed survey on the q-classical orthogonal polynomials of the Hahn class is presented.
Such polynomials appear to be the bounded solutions of the so called q-hypergeometric difference equation
having polynomial coefficients of degree at most two. The central idea behind our study is to discuss in
a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference
equation by means of a qualitative analysis of the relevant q-Pearson equation. To be more specific, a
geometrical approach has been used by taking into account every possible rational form of the polynomial
coefficients, together with various relative positions of their zeros, in the q-Pearson equation to describe
a desired q-weight function on a suitable orthogonality interval. Therefore, our method differs from the
standard ones which are based on the Favard theorem and the three-term recurrence relation.
** This is a joint work with: R. Álvarez-Nodarse and H. Ta?eli.
T. Koornwinder, University of Amsterdam, Korteweg-de Vries Institute, Netherlands
On limits in the q-Askey scheme
Abstract
I will discuss some limit formulas in the (q−)Askey scheme, as given by Koekoek, Lesky & Swarttouw, and
give more conceptual alternatives for them, such that one has really a limit of orthogonal polynomials. This
concerns in particular the limit from q−Racah polynomials to big q−Jacobi polynomials. Where possible,
I will give an interpretation of limit formulas in (quantum) groups.
F. van de Bult, California Institute of Technology, USA
The q-Askey scheme and beyond: q-hypergeometric limits of elliptic
hypergeometric biorthogonal functions
Abstract
26
CHAPTER 4. TUESDAY, AUGUST 30
Elliptic hypergeometric functions are a relatively new (circa 10 years old) generalization of the q-hypergeometric
functions, involving a new parameter p. Letting p → 0 gives limits from elliptic hypergeometric functions
to q-hypergeometric functions. The precise limit depends on how the remaining parameters behave as
p → 0. In this talk I will discuss the different limits one can obtain when taking limits of the elliptic hypergeometric biorthogonal functions first discovered by Spiridonov and Zhedanov. This gives us many families
of biorthogonal q-hypergeometric rational functions, several of which appear to be new. Of special interest
is the case when this biorthogonality reduces to orthogonality between polynomials, which corresponds
exactly to the q-Askey scheme.
The same considerations also work for the limits of the BCn -symmetric multivariate extension of
the elliptic hypergeometric biorthogonal functions by Rains. The multivariate biorthogonal families of
q-hypergeometric functions thus obtained include the Koornwinder polynomials and the Macdonald polynomials.
** This is a joint work with: E. Rains.
M. Foupouagnigni, University of Yaounde I, Cameroon
On Hahn Problem for Orthogonal Polynomials of quadratic and q-quadratic
lattices
Abstract
Following work done by Hahn for orthogonal polynomials of a continuous variable, we prove that any family
of orthogonal polynomials satisfying a second-order difference, q-difference or divided-difference equation
is semi-classical, ie, to say that it’s corresponding functional satisfies a Pearson-type equation.
** This is a joint work with: S. Mboutngam.
L. Vinet, Université de Montréal, Canada
Dunkl shift operators and Bannai-Ito polynomials
Abstract
We consider Dunkl shift operator L with the following properties: (i) L is of first order in the shift and
the reflection operators; (ii) L preserves the space of polynomials of a given degree; (iii) L is potentially
self-adjoint. We show that under these conditions, the operator L has eigenfunctions which coincide with
the Bannai-Ito polynomials. We construct a polynomial basis which is lower-triangular and two-diagonal
with respect to the action of the operator L. This allows to express the BI polynomials explicitly. We
also present an anti-commutator AW(3) algebra corresponding to this operator. From the representations
of this algebra we derive the structure and recurrence relations of the BI polynomials. We introduce new
orthogonal polynomials - referred to as the complementary BI polynomials - as an alternative q → −1 limit
of the Askey-Wilson polynomials. These complementary BI polynomials lead to a new explicit expression
for the BI polynomials in terms of the ordinary Wilson polynomials.
4.3. SHORT COMMUNICATIONS
27
4.3.3 Room C (4.1.E03)
P. Sánchez-Moreno, University of Granada, Spain
Renyi entropy and linearization of orthogonal polynomials
Abstract
The quantification of the spreading of the classical orthogonal polynomials pn (x) is investigated by means
of the Rényi entropy of the associated Rakhmanov probability densities, ρ(x) = ω(x)p2n (x), where ω(x) is
the corresponding weight function. The Rényi entropy is closely related to the Lq -norm of the polynomials.
It is calculated by use of some scarcely known linearization formulas of various types, which make use of
generalized multivariate hypergeometric functions. Explicit applications for Hermite, Laguerre and Jacobi
polynomials are presented and numerically discussed. Finally, the asymptotics of the Lq norms of Hermite
polynomials will be discussed.
1 P. Sánchez-Moreno, D. Manzano, J.S. Dehesa, Direct spreading measures of Laguerre polynomials,
J. Comput. Appl. Math. 235 (2011) 1129-1140.
2 P. Sánchez-Moreno, A. Zarzo, J.S. Dehesa, Rényi entropy and linearization of orthogonal polynomials.
Preprint (2011).
3 A.I. Aptekarev, J.S. Dehesa, P. Sánchez-Moreno and D.N. Tulyakov, Asymptotics lf L p -norms of
Hermite polynomials and Rényi entropy of Rydberg oscillator states. Preprint 2011.
** This is a joint work with: A. I. Aptekarev, J. S. Dehesa D. N. Tulyakov and A. Zarzo.
G. H. Kim, KangNam University, South Korea
Superstability of the Lobacevski equation in normed algebras
Abstract
The aim of this paper is to investigate the superstability of the Lobacevski equation
f
x + y 2
= f (x) f (y)
2
in unital normed algebras.
** This is a joint work with: C. Park.
C. Park, Hanyang University, South Korea
Approximation of an orthogonally additive-quadratic functional equation in
orthogonality spaces
Abstract
28
CHAPTER 4. TUESDAY, AUGUST 30
Using the fixed point method, we prove the Hyers-Ulam stability of the orthogonally additive-quadratic
functional equation
x+y+z
x+y−z
x−y+z
y+z−x
f
+f
+f
+f
2
2
2
2
= f (x) + f (y) + f (z)
for all x, y, z with x ⊥ y, where ⊥ is the orthogonality in the sense of Rätz.
** This is a joint work with: G. H. Kim, J. R. Lee, D. Y. Shin.
E. Berdysheva, University of Hohenheim, Germany
On Turán’s problem for `1 -radial, positive definite functions
Abstract
We consider the following problem: given a convex closed set Ω in Rd , symmetric about
0, and a continuous,
R
positive definite, integrable function f with supp f ⊆ Ω and f (0) = 1, how large can Ω f be? In a particular
case, this problem was posed in the 1970s by P. Turán in a discussion with S. B. Stechkin in connection
with applications in number theory. It turned out that special cases of this problem were, in fact, considered
earlier: in the 1930s by C. L. Siegel in connection with the Minkowski Lattice Point Theorem and in the
1940s by R. P. Boas, Jr. and M. Kac. Turán’s problem and its variants have applications in the study of
sphere packings, in additive number theory, in the theory of entire functions, and other areas of mathematics
as well as in optics, radar engineering, and statistics.
In this talk we discuss the specialization of Turán’s problem to classes of `1 -radial functions (i.e., when
the value of the function depends only on the `1 -norm of the argument). In the latter case, Fourier analysis
of `1 -radial functions can be developed analogously to the classical `2 -theory due to I. J. Schoenberg. The
`1 -theory, however, is more complicated. It was basically developed by H. Berens, Y. Xu, and W. zu Castell,
among others, and leads to a pair of integral transforms on the half-line that can be connected to the Hankel
transform.
To estimate the value of Turán’s problem for `1 -radial functions, we consider its discrete analog.
Investigations on the discrete problem lead to the study of certain monotonicity properties of spherical
Bessel functions. We present a solution for dimensions 2, 3 and 5 and formulate a general conjecture.
** This is a joint work with: H. Berens.
J. Griffin, American University of Sharjah, United Arab Emirates
Moment Sequences related to the Akhiezer Polynomials
Abstract
The construction of some moment sequences on several intervals will be presented. For the simplest case
the sequence is evaluated explicitly. Possible applications relating to special functions will be discussed.
4.3. SHORT COMMUNICATIONS
29
O. Ciaurri, Universidad de La Rioja, Spain
Transplantation for Fourier-Neumann expansions
Abstract
Let Jν be the Bessel function of order ν. The system of functions {φnα }n≥0 , where
p
φnα (x) = 2(α + 2n + 1)x−1/2 Jα+2n+1 (x),
is an orthonormal system in L2 (0, ∞) for α > −1. Moreover, the system is complete in an adequate subspace
of L p (0, ∞). The Fourier series related to this system are known as Fourier-Neumann expansions and they
are defined, for an approprate function f , as
∞
f∼
∑ aαn ( f )φnα ,
with aαn ( f ) =
Z ∞
n=0
0
f (t)φnα (t) dt.
β
In our talk, we will analyze the transplantation operator, Tα , associated to Fourier-Neumann expansions.
For α, β > −1, this operator is defined by
∞
β
Tα f =
∑ aαn ( f )φnβ .
n=0
In particular, we will prove some inequalities in L p (0, ∞) spaces.
R. Vidunas, Kobe University, Japan
Formulas and invariants for tetrahedral Gauss hypergeometric functions
Abstract
Algebraic Gauss hypergeometric functions have a cyclic, a dihedral, the tetrahedral, the octahedral or the
icosahedral projective monodromy groups. One way to give them explicitly is to pullback their differential
equations to a cyclic monodromy group. In the case of the tetrahedral monodromy group, 4 is the minimal
degree of such a pullback transformation. We will express the pullbacked algebraic solutions in terms
of terminating multiple hypergeometric sums. Also expressions for monodromy invariants as terminating
multiple hypergeometric sums will be given.
4.3.4 Room D (4.1.E04)
B. Simanek, California Institute of Technology, USA
Asymptotic Properties of Polynomials Orthogonal over Analytic Regions
Abstract
30
CHAPTER 4. TUESDAY, AUGUST 30
Given a simply connected and bounded region G with analytic boundary, we study the orthogonal polynomials whose measure of orthogonality is carried by G. For a very general class of measures, we will
present several results describing the asymptotic behavior of the orthogonal polynomials everywhere in the
complex plane and the asymptotic behavior of their zeros. We will see many parallels between our results
and the existing theory of orthogonal polynomials on the unit circle. Our results build upon substantial
recent progress in this field and generalize several previously known results.
J. S. Christiansen, University of Copenhagen, Denmark
Orthogonal polynomials on finite and infinite gap sets
Abstract
The theory of orthogonal polynomials on finite unions of compact intervals can be generalized to infinite
gap sets E of Parreau–Widom type. This notion of regular compact sets includes Cantor sets of positive
measure, among others.
In the talk, I will mainly discuss polynomial asymptotics. In particular, we shall investigate to which
extent the orthonormal polynomials Pn admit a power asymptotic behavior (aka Szegő asymptotics). In
this connection, the isospectral torus TE of dimension equal to the number of gaps in E will be introduced
as the key player. Our analysis relies on the covering space formalism introduced into spectral theory by
Sodin–Yuditskii. This allows us to ‘lift’ functions on the multiply connected domain C \ E to the unit disk
D. The universal covering map x : D → C \ E is linked with a Fuchsian group Γ of Möbius transformations
in such a way that
x(z) = x(w) ⇐⇒ ∃γ ∈ Γ : z = γ(w).
As we shall see, it is crucial whether or not the Abel map from TE to Γ∗ , the multiplicative group of
characters of Γ, is a homeomorphism.
M. Lukic, California Institute of Technology, USA
Orthogonal polynomials with recursion coefficients of generalized bounded
variation
Abstract
We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients
satisfying an ` p condition and a generalized bounded variation condition. This latter condition requires
that a sequence can be expressed as a sum of sequences β (l) , each of which has rotated bounded variation,
(l)
iφl (l)
i.e., ∑∞
n=0 |e βn+1 − βn | < ∞.
We prove absence of singular continuous spectrum, preservation of absolutely continuous spectrum
from the corresponding free case, and that pure points embedded in the continuous spectrum can only
occur in an explicit finite set.
More information can be found at http://arxiv.org/abs/1008.3844.
4.3. SHORT COMMUNICATIONS
31
R. Sfaxi, Faculté des Sciences de Gabès, Tunisia
On positive definiteness of some linear functionals with semiclassical
character
Abstract
In this paper we investigate the positive definiteness of the linear functionals ΦU and θc (Φ)U , for any given
semiclassical positive definite linear functional U defined in the linear space of polynomials in one variable
with real coefficients, solution of Pearson equation: (ΦU )0 + ΨU = 0, where Φ and Ψ are polynomials
with arbitrary degree and c is any root of Φ. Some relations between the corresponding sequences of monic
orthogonal polynomials are presented.
** This is a joint work with: W. Chammam and F. Marcellán.
S. Kamioka, Kyoto University, Japan
Orthogonal functions, continued fractions and walks on graphs
Abstract
Combinatorial aspects of orthogonal polynomials and other classes of orthogonal functions have been found
and investigated in relation with various enumerative problems of combinatorial objects. In this talk, I will
focus on walks on lattice or lattice-like graphs (square, triangular, ...), and characterize several classes
of orthogonal functions in a combinatorial way. Such a combinatorial approach in terms of walks (or
path diagrams) goes back to the studies around 1980, especially, Viennot’s combinatorial works on general
orthogonal polynomials (OPs) and Flajolet’s combinatorial interpretation of continued fractions. I will show
that the combinatorial techniques used for OPs and continued fractions can be modified and extended for
wider classes of orthogonal functions and continued fractions, including matrix or vector OPs, Laurent
biorthogonal polynomials, Thiele fractions and so on. As an application, I will also discuss connections
with vicious walks and integrable systems.
M. M. Rodrigues, University of Aveiro, Portugal
Wave solution of two-parameter fractional Schrödinger equation.
Abstract
This work is intended to investigate the multi-dimensional space-time fractional Schrödinger equation of
the form
ih̄ C β Dtα+ u (t, x) =
∇ u (t, x)
0
2m
with h̄ the Planck’s constant divided by 2π, m is the mass and u(t, x) is a wave function of the particle.
Here C Dtα+ ,C ∇β are operators of the Caputo fractional derivatives, where 0.
C
0
** This is a joint work with: S. Yakubovich and N. Vieira.
32
CHAPTER 4. TUESDAY, AUGUST 30
Y. Grandati, Université Paul Verlaine-Metz, France
Rational extensions of solvable potentials and exceptional orthogonal
polynomials.
Abstract
A new approach is developed to generate rational solvable extensions of translationally shape invariant potentials. It uses Darboux-Bäcklund transformations built from unphysical eigenfunctions which are selected
via the symmetry properties of the initial potential. The exceptional orthogonal polynomials underlie the
eigenstates of some extended hamiltonians generated in this way.
4.4 Posters
1. M. Pilar Alfaro, Universidad de Zaragoza, Spain.
On the zeros of orthogonal polynomials on the unit circle. A dynamical view.
Abstract: Let {zn } be a sequence in the unit disk D = {z ∈ C : |z| < 1}. It is known that there exists
a unique positive Borel measure µ in the unit circle Γ = {z ∈ C : |z| = 1} such that the family of
orthogonal polynomials {Φn } associated to µ satisfy
Φn (zn ) = 0,
n=1,2,. . .
We study some features of the measure µ and the orthogonal polynomials Φn in terms of the sequence
{zn }. For example, the asymptotic distribution of the zeros of such orthogonal polynomial is given
in terms of the asymptotic behavior of the sequence {zn }.
References.
• M.P. Alfaro, M. Bello Hernández, J.M. Montaner and J.L. Varona, Some asymptotic properties
for orthogonal polynomials with respect to varying measures. J. Approx. Theory 135 (2005),
22–34.
• M.P. Alfaro and L. Vigil, Solution of a problem of P. Turán on zeros of orthogonal polynomials
on the unit circle. J. Approx. Theory 53 (1988), 195–197.
• A. Martı́nez-Finkelshtein; K.T.-R. McLaughlin and E.B. Saff, Szegő orthogonal polynomials
with respect to an analytic weight: canonical representation and strong asymptotics. Constr.
Approx. 24 (2006), 319–363.
• H.N. Mhaskar and E.B. Saff, On the distribution of zeros of polynomials orthogonal on the
unit circle, J. Approx. Theory 63 (1990), 30–38.
• P. Nevai and V. Totik, Orthogonal polynomials and their zeros, Acta Scient. Math. (Szeged)
53 (1989), 99–104.
• B. Simon, Fine structure of the zeros of orthogonal polynomials. I. A tale of two pictures.
Electron. Trans. Numer. Anal. 25 (2006), 328–368.
**This is a joint work with: M. Bello, J. M. Montaner.
2.
4.4. POSTERS
33
3. Jung Rye Lee, Daejin University, South Korea.
Orthogonal stability of a cubic-quadratic functional equation.
Abstract: Using the fixed point method, we prove the Hyers-Ulam stability of the orthogonally cubicquadratic functional equation
f (2x + y) + f (2x − y) = 2 f (x + y) + 2 f (x − y)
+ 8 f (x) − 4 f (−x) − f (y) − f (−y)
for all x, y with x ⊥ y, where ⊥ is the orthogonality in the sense of Rätz.
4. D. Batista, San Diego State University -LBNL, USA.
Mimetic approximations on non-uniform meshes.
Abstract: Mimetic operators or summation-by-part operators are approximations that satisfy discrete
versions of continuum conservation laws. In 2003 J.E. Castillo and R.D. Grone developed a way
of constructing high order gradient and divergence approximations with mimetic properties for one
dimensional problems on uniform grids. The main attribute of Castillo-Grone operators are that
they preserve symmetry properties of the continuum, they have an overall high order accuracy, and
no numerical artifacts such as ghost points or extended grids are used in their formulation. In this
work we show a generalization of Castillo-Grone schemes for non-uniform, one and two dimensional
meshes. The technique is based on the application of local transformations and the new operators
preserve the valuable attributes of the original uniform ones. We used the non-uniform mimetic
operators to solve boundary-layer like problems on non-uniform meshes. Numerical results show that
the implementation of the new schemes along with adaptive meshes maintains the same order of
accuracy as the Castillo-Grone uniform operators, while decreasing the convergence-rate constant.
5. Manuel Bello-Hernández, Universidad de la Rioja, Spain.
Convergence Rate of Fourier-Padé Approximants for Stieltjes Functions.
Abstract: We obtain rate of convergence of diagonal multipoint Padé approximants of Stieltjes functions. This is used for the study of the convergence of Fourier-Padé and nonlinear Fourier-Padé
approximants for such type of functions. vspace0.25cm
References.
•
M. Bello-Hernández, J. Mı́nguez-Ceniceros (2006), Convergence of Fourier-Padé approximants
for Stieltjes functions. Canad. J. Math. 58, no. 2, 249–261.
• G. López-Lagomasino, A. Martı́nez-Finkelshtein (1995), Rate of convergence of two point
Padé approximants and logarithmic asymptotics of Laurent-type orthogonal polynomials, Constr.
Approx. 11, 255–286.
• A. A. Gonchar, E. A. Rakhamanov, S. P. Suetin (1990), On the rate of convergence of Padé approximants of orthogonal expansions. “Progress in Approximation Theory (Tampa, FL, 1990),”
169–190, Springer Ser. Comput. Math., 19, Springer, New York, 1992.
• H. Stahl, V. Totik (1992), General Orthogonal Polynomials. Encyclopedia of Mathematics and
its Application, vol. 43. Cambridge University Press.
34
CHAPTER 4. TUESDAY, AUGUST 30
6. J. Cacoq, Universidad Carlos III de Madrid, Spain.
Some Results on the convergence of rows of simultaneous Padé approximation.
Abstract: In this poster we present results on the convergence of rows of simultaneous Padé approximation which extend previous ones of P.R. Graves -Moris and E.B. Staff.
7. Giovanni A. Cassatella-Contra, Universidad Complutense de Madrid, Spain.
Some Riemann–Hilbert problems, matrix orthogonal polynomials and discrete matrix equations
with singularity confinement
Abstract: Matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert
problem. This approach[7] provides an easy derivation of discrete equations for the corresponding
matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case,
associated with matrix quartic potentials. It is shown that, when the initial condition and the measure
are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement
property, independently if the solution under consideration is given by the recursion coefficients to
quartic Freud matrix orthogonal polynomials or not.
References.
• G. A. Cassatella-Contra, M. Mañas, Riemann–Hilbert problems, matrix orthogonal polynomials
and discrete matrix equations with singularity confinement, arXiv: 1106.0036v1.
8. K. Castillo, Universidad Carlos III de Madrid, Spain.
Generators of the group of rational spectral transformations for nontrivial C -functions.
Abstract: In this talk we consider transformations of sequences of orthogonal polynomials associated
with a Hermitian linear functional L using spectral transformations of the corresponding C -function
FL . We show that a rational spectral transformation of FL with Laurent polynomial coefficients is
a finite composition of four fundamental elementary spectral transformations.
Keyword: Orthogonal polynomials; linear functional; Stieltjes functions; C -functions; spectral transformations
MSC 42C05
** This is a joint work with: F. Marcellán.
9. A. Cachafeiro, Universidad de Vigo, Spain.
Lagrange interpolation on the circle. Nodal systems and convergence
Abstract We study the convergence of the Laurent polynomials of Lagrange interpolation on the
unit circle for continuous functions satisfying a condition about their modulus of continuity. The
novelty of the result is that now the nodal systems are more general than those constituted by
the n roots of complex unimodular numbers and the class of functions is wider than the usually
studied. Moreover, some consequences for the Lagrange interpolation on [−1, 1] and the Lagrange
trigonometric interpolation are obtained.
** This is a joint work with: E. Berriochoa and J. Garcı́a Amor.
4.4. POSTERS
35
10. J. A. Borrego Morell, Universidad Carlos III de Madrid, Spain.
Orthogonality with respect to a Laguerre differential operator and a fluid dynamical model.
Abstract: Let µ be a finite positive Borel measure on R and L [ f ] = f 00 + (α + 1 − x) f 0 with α > −1.
We study algebraic, analytic and asymptotic properties of the sequence of monic polynomials {Qn }
that satisfy the orthogonality relations
Z
R
L [Qn ](x)xk dµ(x) = 0 for all
0 ≤ k ≤ n−1
A fluid dynamics model for source points location of a flow of an incompressible fluid with preassigned
stagnation points is also considered.
5
Wednesday, August 31
5.1 Schedule of the day
Chair of plenary session
09:30 to 10:20
10:30 to 11:20
Christian Berg
Plenary lecture by Barry Simon
Plenary lecture by Ed. B. Saff
11:20 to 12:00
Room
Chair of session
12:00 to 12:25
12:30 to 12:55
13:00 to 13:25
13:25 to 15:00
15:00 to 2x:00
Poster session II(∗) and coffee break
A (Auditorium)
B (4.1.E01)
C (4.1.E03)
D (4.1.E04)
Theodore Chihara
Javier Segura
A. Sri Ranga Alicia Cachafeiro
Jeff Geronimo
Mohamed Atia
Lin Yu
Burcin Oktay
Mikhail Tyaglov
Luis Garza
Kei Fung Lee
Maria Rebocho
Wadim Zudilin
Roberto Costas
Dan Betea
Darren Ong
Lunch
Excursion to Segovia and Conference dinner
(∗)
1.2.3.4.5.-
The following posters will be presented during the Poster Session of the day.
Alfredo Deaño Cabrera.
6.- Ahmed Fitouhi.
Branislav Dobrucky.
7.- José Gascón.
Paulo Enrique Fernández Moncada. 8.- Edmundo J. Huertas Cejudo.
Héctor Raúl Fernández Morales.
9.- A. Kononova.
Chelo Ferreira.
5.2 Plenary lectures
Barry Simon, Califormia Institute of Technology, USA.
Title: Cantor Polynomials and Their Brothers.
Abstract
By Cantor Polynomials, we mean the OPs associated to the classical Cantor measure (translated to be
symmetric about zero). We show convincing numerical evidence that the Jacobi parameters are asymptotically almost periodic. We consider three other families with singular continuous spectrum of zero Lebesgue
measure and likely fractal structure - critical almost Mathieu, Fibonacci and Doubling Julia - and discuss
many open problem problems and a few results for these classes.
This is joint work with Helge Krueger.
37
38
CHAPTER 5. WEDNESDAY, AUGUST 31
Ed Saff, Vanderbilt University, USA.
Title: Discretizing compact manifolds with minimum energy.
Abstract
he problem of finding configurations of points that are optimally-distributed on a set appears in a number
of guises including best-packing problems, coding theory, geometrical modeling, statistical sampling, radial
basis approximation and golf-ball design (i.e., where to put the dimples).
This talk will focus on classical and recent results concerning geometrical properties of N-point configurations {xi }Ni=1 on a compact metric set A (with metric m) that minimize a weighted Riesz s-energy
functional of the form
w(xi , x j )
∑ m(xi , x j )s ,
i6= j
for a given ‘weight’ function w on A × A and a parameter s > 0.
Specifically, if A supports an (Ahlfors) α-regular measure µ, we prove that whenever s > α, any sequence
of weighted minimal Riesz s-energy N-point configurations on A (for ‘nice’ weights) is quasi-uniform in the
sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore,
if A is an α-rectifiable compact subset of Euclidean space with positive and finite α-dimensional Hausdorff
measure, one may choose the weight w to generate a quasi-uniform sequence of configurations that also
has (as N → ∞) a prescribed positive continuous limit distribution with respect to α-dimensional Hausdorff
measure.
This is joint work with S. Borodachov, D. Hardin and T. Whitehouse.
1. S.V. Borodachov, D.P. Hardin, E.B. Saff, Asymptotics for discrete weighted Riesz energy problems
on rectifiable sets, Trans. Amer. Math. Soc. 360 (2008), 1559-1580.
2. D. P. Hardin, E. B. Saff, and J. T. Whitehouse, Quasi-uniformity of Minimal Weighted Energy Points
on Compact Metric Spaces, arXiv 1104.2911 (2011).
5.3 Short communications
5.3.1 Room A (Auditorium)
J. Geronimo, Georgia Tech, USA
Matrix Orthogonal polynomials and the construction of bivariate real
valued orthogonal polynomials
Abstract
The theory of bivariate real valued orthogonal polynomials constructed using the lexicographical or reverse lexicographical orderings has a close connection with matrix orthogonal polynomials. We will review
this connection and show how the theory of matrix orthogonal polynomials sheds light on the theory of
multivariate orthogonal polynomials.
** This is a joint work with: P. Iliev.
5.3. SHORT COMMUNICATIONS
39
M. Tyaglov, Technische Universität Berlin, Germany
Generalizations of some classical results to matrix-valued
Abstract
I will describe recent generalizations of Szego’s theorem and other asymptotic results to matrix-valued
orthogonal polynomials on the unit circle.
** This is a joint work with: M. Derevyagin, O. Holtz and S. Khrushchev.
W. Zudilin, The University of Newcastle, Australia
Generating functions of Legendre polynomials
Abstract
60 years ago, Fred Brafman derived several “unusual” generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials Pn (x). His result was a consequence of Bailey’s identity for a
special case of Appell’s hypergeometric function of the fourth type. In my talk I present a generalisation
of Bailey’s identity and its implication to generating functions of Legendre polynomials of the form
∞
∑ un Pn (x)zn ,
n=0
where un is an Apéry-like sequence, that is, a sequence satisfying
(n + 1)2 un+1 = (an2 + an + b)un − cn2 un−1
for n = 0, 1, 2, . . . ,
u−1 = 0, u0 = 1,
for a given data a, b and c. Using both Brafman’s generating function and our new results, we construct
a new family of identities for π.
** This is a joint work with: H. H. Chan (National University of Singapore) and J. Wan (University of
Newcastle).
5.3.2 Room B (4.1.E01)
A. Mohamed Jalel, Faculté des sciences de Gabes, Tunisia
An explicit formula for the linearization coefficients of Bessel polynomials
Abstract
Berg and Vignat (Constructive Approximation, 27 (2008), 15-32.) proved positivity results about linearization and connection coefficients for Bessel polynomials. They also obtain some explicit formulas for these
coefficients in some special cases. In this paper, we give an explicit formula of the linearization coefficients
as a single sum and show that this is a generalization of Berg and Vignat’s results.
40
CHAPTER 5. WEDNESDAY, AUGUST 31
L. Garza, Universidad de Colima, Mexico
Perturbations of Laguerre-Hahn class linear functionals by Dirac delta
derivatives
Abstract
In this contribution we analyze perturbations to linear functionals (both on the real line and on the unit
circle) that belong to the Laguerre-Hahn class. In particular, we obtain an expression for the Stieltjes and
Carathéodory functions associated with the perturbed functionals, and we show that the Laguerre-Hahn
class is preserved. We also discuss the invariance of the class under the Szegő transformation.
Keywords: Linear functionals, Laguerre-Hahn class, Stieltjes functions, Carathéodory functions, Szegő
transformations, orthogonal polynomials
2000 AMS classification: 42C05, 33C47
** This is a joint work with: H. Dueñas.
R. S. Costas-Santos, Universidad de Alcala, Spain
Old and New results on Sobolev and semi-classical orthogonal polynomials
Abstract
We present on the one hand algebraic and differential/difference properties for semiclassical-Sobolev polynomial, which are orthogonal with respect to the inner product
hp, riS = h u, p ri + λ h u, Dp Dri ,
where u is a semiclassical linear functional, D is the differential (or the difference or the q–difference,
respectively) operator, and λ is a positive constant; As well as algebraic relations between them and the
polynomial sequence orthogonal with respect to the semiclassical functional u. And on the other hand
we present a degenerate version of the Favard’s Theorem which is valid for all sequences of polynomials
satisfying a three-term recurrence relation
xpn = αn pn+1 + β pn + γn pn−1 ,
even when some coefficient γn vanishes, i.e., the set {n : γn = 0} 6= 0.
/
5.3.3 Room C (4.1.E03)
L. Yu, City University of Hong Kong, China
Global Asymptotic for the Hahn polynomials
Abstract
5.3. SHORT COMMUNICATIONS
41
The Hahn polynomials are a very important class of the discrete orthogonal polynomials arising in various
problems of physics, mathematics and engineering sciences. In the application, we are interested in their
asymptotic behavior when the ratio n/N tends to a constant c ∈ (0, 1) as n → ∞. It has been known that
the existing asymptotic methods for integrals and differential equations are not applicable in the case of the
Hahn polynomials. Using the recently steepest descent method for Riemann-Hilbert problems, we not only
obtain uniform asymptotic formulas for these polynomials, but we also show that the results hold globally
in two regions in the complex z-plane.
** This is a joint work with: R. Wong.
K. Fung Lee, City University of Hong Kong, China
Uniform Asymptotic Expansions of the Tricomi-Carlitz Polynomials
Abstract
The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation
(α)
(α)
(α)
(n + 1) fn+1 (x) − (n + α) x fn (x) + fn−1 (x) = 0,
(α)
n ≥ 1,
(α)
with initial values f0 (x) = 1 and f1 (x) = αx, where x is a real variable and α is a positive parameter. An
asymptotic expansion is derived for these polynomials by using different approaches, including the turningpoint theory for three-term recurrence relations developed by Wang and Wong [Numer. Math. 91(2002)
and 94(2003)], method of steepest descent. Comparisons of these results are made and numerical
√ results
are attached. The result holds uniformly in regions containing the critical values x = ±2/ ν, where
ν = n + 2α − 1/2.
** This is a joint work with: R. Wong.
D. Betea, California Institute of Technology, USA
Lozenge tilings and elliptic special functions
Abstract
We will describe an elliptic distribution on lozenge tilings of a hexagon (boxed plane partitions) that generalizes previous objects studied Random Matrix Theory and Combinatorics to the elliptic hypergeometric
level. An efficient algorithm to sample from the distribution exists and is based on quasi commutation of
difference operators discovered by Rains. The N point correlation function is the discrete Selberg density
(residue of the elliptic Selberg integral). In the scaling limit we exhibit a frozen boundary behavior that is
believed to be new (the frozen boundary can acquire 3 cusps in the hexagon). The study of the boundary
amounts to studying asymptotics of elliptic biorthogonal functions discovered by Spiridonov and Zhedanov;
alternatively one can set up a variational problem a la Kenyon and Okounkov. If time permits, connections
will be made to gap probabilities and the difference elliptic Painlevé equation via tau functions (work in
progress). This talk generalizes results by Borodin, Gorin and Rains and is an application of elliptic special
functions to solvable (dimer) models in statistical mechanics.
The work was done under the supervision of E. Rains and coadvised by A. Borodin.
42
CHAPTER 5. WEDNESDAY, AUGUST 31
5.3.4 Room D (4.1.E04)
B. Oktay, Balikesir University, Turkey
An Approximation Problem over a Subclass of Smooth Domains
Abstract
Let G be a domain from a subclass of smooth domains, φ0 be the Riemann conformal mapping from G
onto a disk with radius r0 , by the conditions φ0 (z0 ) = 0, φ00 (z0 ) = 1, for fixed z0 ∈ G , and let Pn be the
orthogonal polynomials over G. In this work, we investigate the approximation problem of φ0 by elegant
polynomials, called Bieberbach polynomials and admit the representation
Rz n−1
∑ Pj (z0 )Pj (ζ ) dζ
z0 j=0
,
n−1 z∈G
2
∑ Pj (z0 )
j=0
** This is a joint work with: D. M. Israfilov.
M. B. Rebocho, CMUC, Portugal
Structure relations for OPUC
Abstract
In this talk we present a study about orthogonal polynomials on the unit circle, say {Rn }, {Pn }, connected
[1]
by a structure relation ∑sk=0 βn+s,k Rn+s−k = ∑rk=0 αn+s,k Pn+s−k , ∀n ≥ 1 , where the αn ’s and the βn ’s are
[1]
complex numbers, s, r are some non-negative integers, and where Pn+s−k denotes the monic polynomial
0
Pn+s−k
n+s−k .
For certain values of s, r we show the semiclassical character of {Rn } and {Pn }, and we show that the
corresponding linear functionals of orthogonality are a rational transformation of each other.
** This is a joint work with: A. Branquinho.
D. C. Ong, Rice University, USA
Limit-Periodic Verblunsky Coefficients for OPUC
Abstract
Avila recently introduced a new method for the study of the discrete Schrödinger Operator with limit
periodic potential. We adapt this method to the context of orthogonal polynomials in the unit circle
with limit periodic Verblunsky Coefficients. Specifically, we represent these Verblunsky Coefficients as a
continuous sampling of the orbits of a Cantor group by a minimal translation. We then investigate the
measures that arise on the unit circle as we vary the sampling function.
** This is a joint work with: D. Damanik.
5.4. POSTERS
43
5.4 Posters
1. A. Deaño Cabrera, Universidad Carlos III de Madrid, Spain
Orthogonal polynomials for complex Gaussian quadrature
Abstract We consider polynomials which are orthogonal on contours in the complex plane with respect
to exponential weights of the form e−V (z) , where V (z) is a polynomial. When the degree of V (z) is
even, integration on the real axis is feasible, using Hermite–type quadrature. When the degree of
V (z) is odd, it is necessary to consider suitable contours in the complex plane, going to infinity with
the correct phase. For some particular choices of contours, this construction is relevant in the design
of efficient complex quadrature rules for oscillatory integrals on the real axis, via an application of
the classical steepest descent method. We will present recent results and ongoing work in the case
where V (z) is cubic and cubic plus a linear term in z, using the Riemann–Hilbert approach.
** This is a joint work with: Daan Huybrechs and Arno B. J. Kuijlaars.
2. Branislav Dobrucky, University of Zilina, Slovak Republic.
Complex Time Function of Converter Output Quantities as Orthogonal Polynomials and their Analysis.
Abstract: The paper deals with the complex- and discrete Fourier transform which has been considered for both three- and two phase orthogonal voltages and currents of systems. The investigated
systems are power electronic converters supplying alternating current motors. Output voltages of
them are strongly non-harmonic ones, so they must be pulse-modulated due to requested nearly sinusoidal currents with low total harmonic distortion. Modelling and simulation experiment results of
half-bridge matrix converter for both steady- and transient states are given under substitution of the
equivalence scheme of the electric motor by resistive-inductive load and back induced voltage. The
results worked-out in the paper confirm a very good time-waveform of the phase current and results
of analysis can be used for fair power design of the systems. Both of quantities of the converter are
usually handling with real time functions. However, the output quantities of real power electronic
converters can be transformed into complex time functions using Park or Clarke transform, respectively, as vectors rotating in complex Gauss plain. The most advantage of this form of presentation
is ? in case of symmetrical system - that periodicity of the waveforms in complex plain is 2m-times
higher then in real time domain. So, the Fourier analysis, also integral values calculation, can be
done more quickly. Another benefit is possibility of direct using of complex Fourier transform/series
is the speed.
**This is a joint work with: M. Marcokova.
3. Paulo E. Fernández Moncada, Universidad Carlos III de Madrid, Spain.
Analytical Sampling, Lagrange-Type Interpolation Series and de Branges Spaces.
Abstract: The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. This theorem can also be formulated in a more general nonorthogonal setting involving
analytic Kramer kernels which are valued in Hilbert spaces. In this poster we present some new problems: A first one is to characterize the situations when the nonorthogonal sampling formulas can be
expressed as Lagrange-type interpolation series. A necessary and suffcient condition is given in terms
of the so-called zero-removing property. On the other hand, de Branges spaces of entire functions
44
CHAPTER 5. WEDNESDAY, AUGUST 31
satisfy orthogonal sampling formulas which can also be written as Lagrange-type interpolation series.
Finally, we give a characterization of certain Hilbert spaces of entire functions, where a sampling
formula holds, as de Branges spaces.
**This is a joint work with: A. G. Garcı́a and M. A. Hernández-Medina.
4. H. R. Fernández-Morales, Universidad Carlos III de Madrid, Spain
Generalized sampling in L2 (Rd ) shift-invariant subspaces with multiple stable generators
Abstract In order to avoid most of the problems associated with classical Shannon’s sampling theory,
nowadays signals are assumed to belong to some shift-invariant subspace. In this work we consider
a general shift-invariant space VΦ2 of L2 (Rd ) with a set Φ of r stable generators. Besides, in many
common situations the available data of a signal are samples of some filtered versions of the signal
itself taken at a sub-lattice of Zd . This leads to the problem of generalized sampling in shift-invariant
spaces. Assuming that the `2 -norm of the generalized samples of any f ∈ VΦ2 is stable with respect to
the L2 (Rd )-norm of the signal f , we derive frame expansions in the shift-invariant subspace allowing
the recovery of the signals in VΦ2 from the available data. The mathematical technique used here
mimics the Fourier duality technique which works for classical Paley-Wiener spaces.
** This is a joint work with: A. G. Garcı́a and G. Pérez-Villalón.
5. Chelo Ferreira, Universidad de Zaragoza, Spain.
The singular Picard-Lindelof’s theorem.
Abstract: The standard Picard-Lindelöf’s theorem cannot be applied to initial value problems where
the initial conditions are given at a singular point of the differential equation. We generalize the
Picard-Lindelöf’s theorem considering initial conditions given at a singular point, and give a constructive algorithm to approximate the solution of the problem. In the linear case, this technique
may be used to approximate special functions.
**This is a joint work with: J. L. López and E. Pérez Sinusı́a.
6. A. Fitouhi, Faculty of sciences Tunis, Tunisia
On some integral transform in Quantum Calculus
Abstract We study some q-integral transformations such that q-trigonometric, q-Bessel and q-Mellin
transform s and we apply them to solve some q-differential difference equations.
7. J. R. Gascón Márquez, Universidad Nacional Abierta, Venezuela.
Invariance of the spectrum of the infinite complex Toda lattice.
Abstract: It is known that the spectrum of the semiinfinite complex Toda lattices is preserved under
the solution flow for certain Jacobi matrices with bounded bands. We try to extend this result for
Jacobi matrices that have unbounded bands that hold a condition grow at infinity.
a) D. Barrios, J. Gascón.
Spectrum and Generation of Solutions of the Toda Lattice. Discrete Dynamics in Nature and Society Volume 2009 (2009), Article ID 237487, 12 pages
doi:10.1155/2009/237487
5.4. POSTERS
45
b) D. Barrios, R. Hernández. On the relation between the Toda and Volterra lattices. Duke
Math. J. 45, no. 2 (1978), 267–310
c) S. Garcı́a. Conjugation and Clark operators. Contem.Math. 393, (2006) 67-112.
d) P. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure. Appl.
Math., 31 (1968) 467-490.
e) Peherstorfer F. Peherstorfer, On Toda lattices and orthogonal polynomials, J. Comput. Appl.
Math. 133 (2001) 519-534.
**This is a joint work with: D. Barrios Rolanı́a.
8. E. J. Huertas Cejudo, Universidad Carlos III de Madrid, Spain
New Asymptotic Results on Laguerre-Sobolev-type Orthogonal Polynomials
Abstract In this contribution we consider sequences of monic Laguerre-Sobolev-type polynomials
(MOPS in short), orthogonal with respect to the inner product
hp, qiS =
Z ∞
p(x)q(x)xα e−x dx + N p0 (a)q0 (a), α > −1
0
where N ∈ R+ , and a ∈ R− .
Our viewpoint sheds some new light on the asymptotic behavior of this MOPS, considering the perturbation of the Laguerre weight supported on the negative semi-axis of the real line. In such a way,
we state a comparative analysis with the behavior of the standard Laguerre-type polynomials.
The outline of the talk is the following. In the first part we introduce a useful rate of convergence
formula for the ratio of two Laguerre standard MOPS. Next, we obtain the relative outer asymptotic
of these polynomials with respect to the standard Laguerre polynomials. The analogue of the MehlerHeine formula as well as a Plancherel-Rotach formula for the rescaled polynomials are given. Finally,
we analyze the behavior of their zeros in terms of their dependence on N.
a) R. Álvarez-Nodarse, F. Marcellán, and J. Petronilho, WKB Approximation and Krall-type Orthogonal Polynomials, Acta Appl. Math. 54 (1998), 27–58.
b) J. Dini and P. Maroni, La multiplication d’ une forme linéaire par une forme rationnelle. Application aux polynômes de Laguerre-Hahn, Ann. Polon. Math. 52 (1990), 175–185.
c) H. Dueñas, and F. Marcellán, Laguerre-Type orthogonal polynomials. Electrostatic interpretation, Int. J. Pure and Appl. Math. 38( 2007), 345–358.
d) F. Marcellan and P. Maroni, Sur l’ adjonction d’ une masse de Dirac à une forme réguliére art
semi-classique Annal. Mat. Pura ed Appl. CLXII (1992), 1–22.
** This is a joint work with: F. Marcellán and and H. Dueñas.
9. A. A. Kononova, Baltic State Technical University, Russia
On perturbation of ratio asymptotics of orthogonal polynomials associated with a system of curves
and arcs
46
CHAPTER 5. WEDNESDAY, AUGUST 31
Abstract We consider a finite mass points perturbation of a measure supported by a system of curves
and arcs in the complex plane. We study the perturbation of ratio asymptotics of associated monic
orthogonal polynomials. Our main result is related with the description of perturbation where the
ratio asymptotics is not changed.
Let µ be a measure supported by a finite union of complex arcs E = ∪ pj=1 , E j ⊂ C and masspoints
{zk }lk=1 , zk ∈ C \ E. The part of the measure supported by the set E supposed to be a.c. and satisfy
the Szegö condition on E. Consider some transformation µ̂ of this measure obtained from µ by adding
or deleting some masses. Let Qn (z) (Q̂n (z)) be the monic polynomials orthogonal with respect to
µ (µ̂). It is known that the weak asymptotics of orthogonal polynomials is stable under such kind
of perturbation, but the strong asymptotics changes depending on the geometry of support of the
measure. Our main result is related with the description of perturbation where the ratio asymptotics
is not changed. Let ω j (z) be the harmonic measure of E j . We show that if ∑lk=1 ω j (zk ) ≡ 0 (mod
Qn (z)
Q̂n (z)
1), k = 1, . . . , p, then
→ 0 (as n → ∞) uniformly in compact neighbourhoods of
−
Qn+1 (z) Q̂n+1 (z)
infinity. In many cases the converse statement is also true. The work is supported by RFFI grant
10-01-00682.
** This is a joint work with: V. A. Kaliagin.
6
Thursday, September 1
6.1 Schedule of the day
Chair of plenary session
09:30 to 10:20
10:30 to 11:20
11:20 to 12:00
Room
Chair of session
12:00 to 12:25
12:30 to 12:55
13:00 to 13:25
13:25 to 15:00
Chair of plenary session
15:00 to 15:50
Room
Chair of session
16:00 to 16:25
16:30 to 16:55
17:00 to 17:25
17:30 to 17:55
18:00 to 18:25
19:00 to 20:00
(∗)
1.2.3.4.5.-
Aleksandre I. Aptekarev
Plenary lecture by Peter Yuditskii
Plenary lecture by Evguenii Rakhmanov
Poster session III(∗) and coffee break
A (Auditorium)
B (4.1.E01)
C (4.1.E03)
Jacob Christiansen
D. Gómez-Ullate
Renato Álvarez
Chistian Berg
Yamilet Quintana
Helge Krueger
Henrik L. Pedersen
Robert Milson
Patanjali Sharma
Alberto Ibort
Anabela Paiva
Wolfgang Erb
Lunch
Paul Nevai
Plenary lecture by Andrei Martı́nez Finkelshtein
A (Auditorium)
B (4.1.E01)
C (4.1.E03)
Adhemar Bultheel
Tom Koorwinder
Eduardo Godoy
Peter Clarkson
Mohamed Gaied
Luis Verde-Star
Cristophe Smet
Jose Cardoso
Carlos Álvarez
Stamatis Koumandos
Sri Ranga
Mariana Marcokova
M. Domı́nguez
Pablo González-Vera M. Asunción Sastre
Steven Delvaux
Hamza Chaggara
Carlos Dı́az
Closing cocktail at Hotel Tryp Leganés
D (4.1.E04)
B. de la Calle
Perdomo Pio
Jesús Illán
Rafal Nowak
D (4.1.E04)
A. Branquinho
Volha Kushel
Franck Wielonsky
Phillip Offner
Celina Pestano
M. Derevyagin
The following posters will be presented during the Poster Session of the day.
Cristian Arteaga.
6.Taro Nagao.
Fernando Llédo Macau.
7.François Ndayitagije.
Héctor Luna.
8.Dilcia Pérez
Juan José Moreno-Balcázar. 9.Vanessa Pirani
Carmen Suárez Rodrı́guez
10.- Concepción González
47
48
CHAPTER 6. THURSDAY, SEPTEMBER 1
6.2 Plenary lectures
Peter Yuditskii, University of Linz, Austria.
Title: On special conformal mappings in approximation theory.
Abstract
Special conformal mappings have been actively used in approximation theory, for example, they reveal the
structure of the polynomials of the least deviation from zero on a system of intervals and describe the
spectral sets of periodic Jacobi matrices. We recall certain classical results and present new ones dealing
with this approach. In particular, by developing this method we obtained precise asymptotics of the error
En of the best polynomial approximation of sgn(x) on two intervals [−A, −1] ∪ [1, B]:
1
−1/2 −nη ϑ0 2 ({nω1 + ω2 } − ω2 )| τ ,
En = (c + o(1))n
e
ϑ0 12 ({nω1 + ω2 } + ω2 )| τ where ϑ0 (v|τ) is the theta-function and {x} denotes the fractional part of x. The constants τ, c, η, ω1 , ω2
are given explicitly by means of elliptic integrals depending on A and B.
Supported by the Austrian Science Fund FWF, project no: P22025–N18.
Evguenii Rakhmanov, University of South Florida, USA.
Title: Critical measures, S-curves and G-functions.
Abstract
Curves with a special symmetry property (S-property) in a harmonic external field play an important role in
several fields of modern analysis such as approximation theory, orthogonal polynomials and nonlinear PDE.
Such curves are also connected to a number of classical problems in geometric theory of analytic functions
(moduli problems, quadratic differentials and so on).
Equilibrium measures of S-curves are critical measures which connect them to a large class of equilibrium
problems for logarithmic potential. On the other hand, potentials of those measures (g-functions) may be
also defined in terms of certain boundary value problems.
The lecture will review some of above mentioned connections and, in particular, some of the old and
new result on the existence problem for S-curves.
Andrei Martinez Finkelshtein, Universidad de Almerı́a, Spain.
Title: Heine, Hilbert, Pade, Riemann, and Stieltjes: a Nuttall’s work 25 years later.
Abstract
In 1986 J. Nuttall published a paper in Constructive Approximation entitled Asymptotics of Generalized
Jacobi Polynomials, where with his usual insight he studied the behavior of the denominators (”generalized Jacobi polynomials”) and the remainders of the Padé approximants to an algebraic function with 3
branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the
6.3. SHORT COMMUNICATIONS
49
generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they
are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality
relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this talk features also a special appearance by Riemann surfaces, quadratic differentials,
compact sets of minimal capacity, special functions and other characters.
This is joint work with E. A. Rakhmanov and S. P. Suetin.
6.3 Short communications
6.3.1 Room A (Auditorium)
C. Berg, University of Copenhagen, Denmark
On generalized Stieltjes–Wigert polynomials
Abstract
The generalized Stieltjes–Wigert polynomials depending on parameters 0 ≤ p < 1 and 0 < q < 1 are
discussed. By removing the mass at zero of the N-extremal solution concentrated in the zeros of the
D-function from the Nevanlinna parametrization, we obtain a discrete measure µ M which is uniquely
determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials
(PnM ). As noticed by T. Chihara, these polynomials are the shell polynomials corresponding to the maximal
parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as
the parameter sequence corresponding to the generalized Stieltjes–Wigert polynomials, and compute the
value of related continued fractions. The mass points of µ M have been studied in recent papers of Hayman,
Ismail–Zhang and Huber. In the special case of p = q, the maximal parameter sequence is constant and
the determination of µ M and (PnM ) gives an answer to a question posed by T. Chihara in 2001.
** This is a joint work with: Jacob S. Christiansen
H. L. Pedersen, University of Copenhagen - Faculty of Life Sciences, Denmark
A study of the volume of the unit ball in Euclidean space based on complex
methods
Abstract
Properties of the volume Ωn of the unit ball in Rn have been investigated by many authors. In terms of
Euler’s gamma function this volume can be expressed as
Ωn =
π n/2
.
Γ(1 + n/2)
Often results are formulated via the quantity
1/n log n
vn = Ωn
.
50
CHAPTER 6. THURSDAY, SEPTEMBER 1
We prove that {vn+2 } is a Hausdorff moment sequence and in particular decreasing and logarithmically
convex.
The proof is based on properties of the functions
Fa (x) =
log Γ(x + 1)
,
x log(ax)
a > 0.
These functions are extended to the complex plane cut along the negative real axis. We obtain that Fa
maps the upper half plane into itself (and hence is a so-called Pick function) when a ≥ 1.
Other results concern the ratio vn /vn+1 . Alzer found the best constants c and d such that for n ≥ 2,
2
2
ec/n(log n) ≤ vn /vn+1 ≤ ed/n(log n) ,
and he proved the estimates 2/3.
** This is a joint work with: C. Berg.
A. Ibort, Dpt. of Mathematics, Univ. Carlos III de Madrid, Spain
A representation theorem for unbounded orthogonally additive polynomials
in Banach lattices
Abstract
A continuous homogeneous orthogonally–additive polynomial on a Banach lattice X is an homogeneous
continuous polynomial P on X such that P(x + y) = P(x) + P(y) whenever |x| ∧ |y| = 0. It was proved by
Benyamini, Lassalle and Llavona that any such polynomial can be represented by a continuous linear map
on the nth concavification of X (n being the degree of homogeneity of P). In
particular if X is a Köthe
R n
space of functions, then there exists a measurable function ξ such that P( f ) = f ξ dµ. We will show that
a similar representation theorem can be obtained for unbounded positive orthogonally–additive polynomials
P. In fact, it will be shown that given P as before there will exists a continuous linear map representing
P on the nth concavification of an extension X̄P of the Banach lattice X that will depend on the given
polynomial. In particular, if P is a positive quadratic polynomial on a Köthe space of functions and the
extension has finite rank, then it will represented as a Sobolev quadratic form.
** This is a joint work with: J. L. G. Llavona.
P. Clarkson, UK
Painlevé Equations – Nonlinear Special Functions
Abstract
In this talk I shall give an overview of the Painlevé equations, which might be thought of nonlinear special
functions, and discuss some of their properties. Further I shall discuss some of the “Painlevé Challenges”,
i.e. open problems in the field of Painlevé equations.
6.3. SHORT COMMUNICATIONS
51
C. Smet, Katholieke Universiteit Leuven, Belgium
Orthogonal polynomials on a bi-lattice
Abstract
We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the
shifted lattice N + 1 − β . We combine both lattices to obtain the bi-lattice N ∪ (N + 1 − β ) and show that
the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system
of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé
equation.
** This is a joint work with: W. Van Assche.
S. Koumandos, University of Cyprus, Cyprus
Complete monotonicity and related properties of some special functions
Abstract
Γ(x+t) s−t+1
We completely determine the set of s,t > 0 for which the function Ls,t (x) := x − Γ(x+s)
x
is a Bernstein
function, that is Ls,t (x) is positive with completely monotonic derivative on (0, ∞). The complete monotonicity of several closely related functions is also established. Our results can be applied to obtain sharp
estimates for certain trigonometric sums.
** This is a joint work with: M. Lamprecht.
M. Domı́nguez de la Iglesia, Universidad de Sevilla, Spain
Bivariate Markov processes and matrix orthogonality
Abstract
It is well known that some (one dimensional) Markov processes are related with orthogonal polynomials.
Such are the cases of random walks and birth and death processes, if the state space S is taken to be
discrete, and diffusion processes, where the state space S is a real interval. In this talk we will explain with
some examples how the matrix orthogonality plays an important role in order to give an interpretation of
bivariate Markov processes, i.e. Markov processes assuming values in S × {1, 2, . . . , N}, where S ⊂ R and N
is a nonnegative integer. We will focus in the case where both components are dependent and give some
examples where S is discrete or continuous.
52
CHAPTER 6. THURSDAY, SEPTEMBER 1
S. Delvaux, Katholieke Universiteit Leuven, Belgium
Isospectral torus for banded Hessenberg matrices
Abstract
A sequence of orthogonal polynomials (Pn (x))∞
n=0

b0
a∗
 0
J=


0
on the real line gives rise to the well-known Jacobi matrix

a0
0

b1 a1

.. .. 
∗
.
.
.
a1

.. ..
.
.
Let us call J periodic with period r if ak ≡ ak+r and bk ≡ bk+r for all k ≥ 0. In this case the zeros of Pn (x)
for n → ∞ are asymptotically distributed on a set Γ ⊂ R which is the disjoint union of at most r intervals
[?].
Conversely, for any such set Γ ⊂ R, one can ask:
• Is there a periodic Jacobi matrix associated to this set Γ?
• If so, what is the structure of the ’isospectral manifold’ formed by all such Jacobi matrices?
The answer to the first question can be yes or no depending on a nice potential theoretic criterion. For the
second question, the structure of the isospectral manifold turns out to be a torus [1].
In this talk we study the generalization of these questions to the setting of banded Hessenberg matrices,
i.e., we replace the Jacobi matrix J by a matrix which has several non-vanishing diagonals in its lower
triangular part. We discuss the connection with Nikishin systems.
[1] B. Simon, Szegö’s Theorem and its Descendants, Princeton university press, 2010.
** This is a joint work with: A. López-Garcı́a.
6.3.2 Room B (4.1.E01)
Y. Quintana, Universidad Simón Bolı́var, Venezuela
Concerning asymptotic behavior for extremal polynomials associated to
non-diagonal Sobolev norms
Abstract
Let P be the space of polynomials with the non-diagonal Sobolev norm k · kW 1,p (V 2/pµ) , given by
k f kW 1,p (V 2/pµ) :=
Z
p/2
dµ
a p | f | + c p | f | + 2 ℜ(b p f f 0 )
2
0 2
1/p
,
where µ := µ1 + µ2 is a measure such that |b p |2 ≤ (1−ε)a p c p , µ1 -almost everywhere for some fixed extremal
polynomials associated to k · kW 1,p (V 2/pµ) , stating hypothesis on the matrix V 2/p rather than on the diagonal
6.3. SHORT COMMUNICATIONS
53
matrix Λ2/p appearing in its unitary factorization. Using properties of the multiplication operator by the
independent variable as the main tools in order to obtain our results, we complement the study started in
[1] where the case µ = µ2 with 1 ≤ p ≤ 2 was considered with a similar approach.
[1] A. Portilla, Y. Quintana, J. M. Rodrı́guez, E. Touris, Zero location and asymptotic behavior for
extremal polynomials with non-diagonal Sobolev norms, J. Approx. Theory 162 (2010) 2225?-2242.
** This is a joint work with: A. Portilla, J. M. Rodrı́guez and E. Tourı́s.
R. Milson, Dalhousie University, Canada
On Orthogonal Polynomials spanning a non-standard flag
Abstract
We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations
that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical
families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial
flags; the lowest degree polynomial has degree m > 0. In this talk we review the classification of codimension m = 1 exceptional polynomials, and give a novel, compact proof of the fundamental classification
theorem for codimension 1 polynomial flags.
** This is a joint work with: D. Gómez-Ullate and N. Kamran.
A. Paiva, Universidade da Beira Interior, Portugal
Laguerre-Hahn Orthogonal Polynomials on the Real Line-Matrix Study
Abstract
This work has the purpose to present a study forLaguerre-Hahn orthogonal polynomials sequences, on
the real line, in terms of second order vectorial differential equations and matrix Sylvester differential
equations. The starting point is the result established by Magnus concerning to the semi-classical orthogonal
polynomials and applying the method developed by Branquinho and Rebocho. Let u be a Laguerre-Hahn
(1)
regular functional, S the Stieltjes function related to u, {Pn }, {Pn } and {qn } be the sequences of the
corresponding orthogonal polynomials, first order associated polynomials and functions of second kind,
(1)
respectively. Consider the vectorial sequence of orthogonal polynomials {ψn }, ψn = [Pn+1 Pn ]T . Through
0
2
this sequences we show an equivalence between Riccatti differential equation AS = BS +CS +D and second
order differential vectorial equations for {ψn } with matrix coefficients. We also give an analogous result for
sequences of second kind functions that satisfies the second order differential equation with non polynomial
coefficients. Considering the second
order#differential equations we obtained a Sylvester matrix equation
"
(1)
Pn+1 Pn
. The solution of this equation is a matrix obtained by Radon’s
Pn Pn−1
Lemma with help of two differential systems. Some examples of application are given.
** This is a joint work with: A. Branquinho and M. das Neves Rebocho.
AYn0 = BnYn −Yn C where Yn =
54
CHAPTER 6. THURSDAY, SEPTEMBER 1
M. Gaied, ISITCom, Hammam Sousse, Tunisia
q-Dunkl-classical q-Hermite type polynomials
Abstract
Let J be a linear operator defined on polynomials by means of
Jxn = αn xn , n = 0, 1, 2, ...
where αn 6= 0 for all n ≥ 0 and let {Pn }n≥0 be a symmetric sequence given by the explicit representation
[ n2 ]
Pn (x) =
∑ cn,k xn−2k
k=0
where cn,0 6= 0 for all n ≥ 0.
In a previous work, we gave, in terms of αn and cn,k , a necessary and sufficient condition on operator J
and symmetric orthogonal polynomial set {Pn }n≥0 to have also {JPn }n≥0 orthogonal and we discussed some
examples.
In this talk, we provide further examples of couples ({Pn }n≥0 , J) for which {Pn }n≥0 and {JPn }n≥0 are
orthogonal. More precisely, we consider the discrete q-Hermite polynomials I and II and an operator related
to the q-Dunkl operator. That allows us to derive new orthogonal polynomial sets. For which, we state
three term recurrence relations, two-order q-difference equations, functionals ensuring the orthogonality
and other properties.
** This is a joint work with: A. Zaghouani and Y. B. Cheikh.
J. L. Cardoso, Universidade de Trás-os-Montes e Alto Douro, Portugal
Basic Fourier expansions on q-linear grids. Convergence issues.
Abstract
In the last decade, complete orthogonal systems were used to establish basic Fourier expansions. A q-type
Hölder condition for uniform.
A. Sri Ranga, Universidade Estadual Paulista, Brazil
A class of hypergeometric polynomials with zeros on the unit circle:
extremal and orthogonal properties and quadrature formulas
Abstract
6.3. SHORT COMMUNICATIONS
55
Let the complex parameter b be such that Re(b) > −1/2. It has been shown recently that the hypergeometric polynomials Sm (b; z) = 2 F1 (−m, b + 1; b + b̄ + 1; 1 − z), m ≥ 0, are the Szegő polynomials with
respect to the weight function [e−θ ]I m(b) [sin2 (θ /2)]Re(b) .
Assuming Re(b) > 0, here we look at some properties of the hypergeometric polynomials Rm (b; z) =
F
(−m,
b; b + b̄; 1 − z), m ≥ 0 and the associated functions Gm (b; x) = (4z)−m/2 Rm (b; z), m ≥ 0, where
2 1
2x = z1/2 + z−1/2 . The polynomial Rm (b; z) is a particular para-orthogonal polynomial of Sm (b; z). Hence,
all m zeros of Rm (b; z) are distinct and lie on the unit circle |z| = 1 and that the function Gm (b; x) has exactly
m simple zeros in [−1, 1]. Results on extremal and orthogonal properties and also quadrature formulas are
obtained.
** This is a joint work with: D. K. Dimitrov and M. E. H. Ismail.
P. González-Vera, La Laguna University, Spain
Quadratures and orthogonality associated with the Cayley transform
Abstract
In this talk, we are dealing with the approximate calculation of the weighted integral on the whole real line,
Z ∞
Iσ ( f ) =
f (x)σ (x)dx
(6.1)
−∞
by means of a quadrature rule
n
Inσ ( f ) =
∑ A j f (x j )
(6.2)
j=1
The method is based upon the pass to the unit circle by means of the so-called “Cayley transform”, that
i−x
mapping the right hand half plane onto the upper half plane and the
is, the bilinear transformation z = i+x
whole real line R onto the unit circle T = {z ∈ C : |z| = 1}. Making use of the Cayley transform in (6.1), it
follows
Z π
Iσ ( f ) = Iω (g) =
g(θ )ω(θ )dθ
(6.3)
−π
Now, we can approximate (6.3) by means of a Szegő or interpolatory-type quadrature formula on the unit
circle,
n
Inω (g) =
∑ λ j g(θ j ), θ j 6= θk if j 6= k and {θ j }nj=1 ⊂ [−π, π)
(6.4)
j=1
Then, by applying the inverse of the Cayley transform in (6.4) we obtain a quadrature rule like (6.2) with
λj
1−zj
Aj = ;
xj = i
; z j = eiθ j , j = 1, . . . , n,
2
1+zj
exactly integrating certain rational functions with prescribed nodes at ±i. Some properties concerning
orthogonality,maximal domains of validity of the quadratures and connections with certain orthogonal
rational functions are presented. Finally, some numerical experiments are also carried out. ** This is a
joint work with: Francisco Perdomo-Pı́o and Michael Stessin.
56
CHAPTER 6. THURSDAY, SEPTEMBER 1
H. Chaggara, Ecole Supérieure des Sciences et de Tech. de Sousse, Tunisia
On Apostol-Euler and Apostol-Bernoulli Polynomials
Abstract
In this talk, we use quasi-monomiality principle to derive some properties and expansion formulas related
to Apostol-Euler and Apostol-Bernoulli polynomials.
6.3.3 Room C (4.1.E03)
H. Krueger, Caltech, USA
Must measures supported on a Cantor set have simple recursion coefficients?
Abstract
Instead of dicussion the question of the title, I will focus on the dual problem. Consider the class of
recursion coefficients given by a(n) = 1 and b(n) = λ f (αnρ ) where λ > 0, α is irrational, ρ > 0, and f is a
1-periodic function, i.e. f (x) = f (x + 1). I will argue these recursion coefficients are only ”simple” when
ρ = 1, respectively when they are almost-periodic.
Then I will discuss my recent progress on showing that in the other case the spectrum of the associated
Jacobi operator is an interval.
P. Sharma, CIITM, Jaipur, India
On the interlacing inequalities for sign-symmetic matrices
Abstract
If λ1 ≥ λ2 ≥ . . . ≥ λn are eigenvalues of a sign-symmetric matrix, and µ1 ≥ µ2 ≥ . . . ≥ µn−1 are eigenvalues
of its principal submatrix, then we prove results concerning the interlacing of eigenvalues of principal
submatrices of sign-symmetric matrices.
** This is a joint work with: O. Y. Kushel.
W. Erb, University of Luebeck, Germany
An orthogonal polynomial analogue of the Landau-Pollak-Slepian
time-frequency analysis
Abstract
6.3. SHORT COMMUNICATIONS
57
The aim of this talk is to present a time-frequency theory for orthogonal polynomials on the interval [-1,1]
that runs parallel to the time-frequency analysis of bandlimited functions developed by Landau, Pollak and
Slepian. For this purpose, the spectral decomposition of a particular compact time-frequency-operator is
studied. This decomposition and its eigenvalues are closely related to the theory of orthogonal polynomials.
Results from both theories, the theory of orthogonal polynomials and the Landau-Pollak-Slepian theory, can
be used to prove localization and approximation properties of the corresponding eigenfunctions. Finally, an
uncertainty principle is shown that reflects the limitation of coupled time and frequency locatability.
L. Verde-Star, Universidad Autonoma Metropolitana, Mexico
Dual polynomial bases, sequences of interpolatory type and almost
orthogonal generalized Vandermonde matrices
Abstract
We use divided differences to construct bases for the dual spaces of vector spaces of polynomials. We also
construct families of polynomial sequences that generalize the Chebyshev family and show how they yield
simple interpolation formulas and almost orthogonal generalized Vandermonde matrices. Representation
of the polynomials as characteristic polynomials of Toeplitz-Hessenberg matrices are also studied.
C. Álvarez Fernández, Universidad Pontificia Comillas, Spain
Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization
and the multi-component 2D Toda hierarchy
Abstract
Multiple orthogonality is considered in the realm of a Gauss-Borel factorization problem for a semiinfinite
moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of
M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality
has a unique solution, that can be obtained from the solution of a Gauss-Borel factorization problem for a
semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that
are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the
moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix
with snake shape, and results like the ABC theorem or the Christoffel-Darboux formula are re-derived in
this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix).
The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss-Borel factorization problem, is discussed.
Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with
solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov-Shabat matrices as well as wave functions and their adjoints are determined. The
58
CHAPTER 6. THURSDAY, SEPTEMBER 1
construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the τ -function
representation of the multiple orthogonality is given.
** This is a joint work with: U. Fidalgo Prieto and M. Mañas.
M. Marcokova, University of Zilina, Slovak Republic
Korous’ theorem revisited
Abstract
105 years ago the czech mathematician Josef Korous was born (1906-1981). This is the opportunity to
remind one of his most important results reached in the theory of orthogonal polynomials which is in
the literature called the Korous’ theorem. It contains a deep result on the upper bounds of polynomials
e
e
{pn (x)}∞
, where w(x)
is
n=0 orthogonal in the interval ha, bi with the weight function w(x) = w(x)k(x)
∞
the weight function of another system of polynomials { pen (x)}n=0 orthogonal in the same interval and
k(x) > k > 0 . We outline some generalizations of this theorem given by several authors. We also take into
account some classes of weights considered in the theory of orthogonal polynomials, especially the ones
used in Korous’ works and the ones occuring in contemporary literature on orthogonal polynomials.
M. A. Sastre Rosa, Universidad Politécnica de Madrid, Spain
Asymptotically Toeplitz Hessenberg operators and the Riemann mapping
Abstract
In a recent work the authors have established a relation between the limits of the elements of the diagonals
of the Hessenberg matrix D associated with a regular measure µ, whenever those limits exist, and the
coefficients of the Laurent series expansion of the Riemann mapping φ (z) of the support supp(µ), when
this is a Jordan arc or a connected finite union of Jordan arcs in the complex plane C.
We extend here this result using asymptotic Toeplitz operator properties of the Hessenberg matrix.
** This is a joint work with: C. Escribano, A. Giraldo and E. Torrano.
C. Dı́az Mendoza, La Laguna University, Spain
Orthogonal Laurent Polynomials revisited
Abstract
Orthogonal Laurent polynomials have been extensively studied in connection with Padé Approximation and
quadrature formulas in the real axis. However, recently, they have become a main tool in the study of
these problems in the Unit Circle. In this talk, a new algebraic approach, with a unified treatment and
6.3. SHORT COMMUNICATIONS
59
some new ways of analysis, is given for general sequences of Orthogonal Laurent Polynomials; in particular,
recurrence relations and Christoffel-Darboux identities play an important role in our analysis.
** This is a joint work with: R. Cruz Barroso and R. Orive.
6.3.4 Room D (4.1.E04)
F. Perdomo-Pı́o, La Laguna University, Spain
Multipoint Padé approximation on the unit circle and the real line
Abstract
Let µ be a measure supported on [−1, 1], its Cauchy Transform,
Z 1
dµ(t)
Fµ (z) =
−1
z−t
,
represents an analytic function on C\[−1, 1]. Fµ (z) is said to be a Markov function.
On the other hand given σ a measure supported on the unit circle T = {z ∈ C : |z| = 1}, its Herglotz-Riesz
Transform,
t +z
dσ (t),
t
T −z
Z
Hσ (z) =
is an analytic function in D = {z ∈ C : |z| related by
Fµ (z) =
z
Hσ (z).
1 − z2
The aim of this talk is to find the connections between certain rational approximants (multipoint Padé
approximants), Fn (x) and Hn (z) to Fµ (x) and Hσ (z) respectively along with their algebraic and analytic
properties when µ and σ are related by (??).
** This is a joint work with: R. Cruz-Barroso and P. González-Vera.
J. Illan, Universidad de Vigo, Spain
A new approach to calculate modified Gauss quadrature formulas
Abstract
Let KW /qdx be an integrator on [−1, 1], where W is the Chebyshev weight function of the first kind, K is
in L2 and q is a polynomial positive on [−1, 1]. We can use the modified Chebyshev algorithm to compute
nodes and coefficients of the Gauss quadrature formula associated with KW /q. If K is an ill-scaled function
then the calculation of the first moments gets more difficult due to numerical instability. To overcome this
drawback we have applied a ”non-standard” formula which is exact for polynomials. In general, this method
works efficiently when dealing with integrals whose mass is unevenly distributed.
** This is a joint work with: E. Berriochoa, A. Cachafeiro and F. Cala.
60
CHAPTER 6. THURSDAY, SEPTEMBER 1
R. Nowak, Institute of Computer Science, University of Wroclaw, Poland
Convergence acceleration of some continued fractions
Abstract
We propose a method of convergence acceleration of continued fractions of the form K(an /bn + a0n /b0n ),
where an , a0n and bn , b0n are polynomials. The method, which is of iterative character, produces tail approximations whose asymptotic expansion’s accuracy is improving in each step. We give examples involving
continued fraction expansions of some mathematical constants, as well as elementary and special functions.
V. Kushel, TU Berlin, Germany
Generalized Brouncker’s continued fractions and their logarithmic
derivatives
Abstract
In this talk, we discuss solutions y(s, r) to the functional equation y(s, r)y(s + 2r, r) = (s + 1)(s + 2r − 1) for
1
r > . They can be expanded into continued fractions generalizing Brouncker’s continued fraction b(s).
2
We obtain formulas for the first and the second logarithmic derivatives of y(s, r). The asymptotic series for
y(s, r) at ∞ are also studied. The generalizations of some Ramanujan’s formulas are presented.
F. Wielonsky, Université de Provence, France
On the convergence of rational interpolants to the exponential function
Abstract
We describe the asymptotic behavior, as n tends to infinity, of rational interpolants p/q, p and q of degree
n, to the exponential function, defined by
2n+1
p(z)e−z/2 + q(z)ez/2 = O( ∏ (z − zi )),
as z → zi ,
i = 1, . . . , 2n + 1.
i=1
In a previous work, locally uniform convergence of the rational interpolants to ez , as n → ∞, was shown
when the complex interpolation points zi are bounded or of logarithmic growth with n. The Deift and Zhou
steepest descent method was the main tool in obtaining this result.
We show that this method can still be used in the case of interpolation points whose modulus grow
like n1−α , 0.
** This is a joint work with: T. Claeys.
6.3. SHORT COMMUNICATIONS
61
Ph. Oeffner, TU Braunschweig, Germany
Spectral Approximation with Appell Polynomials
Abstract
We study the super spectral vanishing viscosity method (SVV) in the context of conservation laws. The
Fourier method may not converge for nonlinear scalar conservation laws which exhibit spontaneous shock
discontinuites. The idea of the super spectral vanishing viscosity method (SSV) is to add some viscosity
term to the equation and approximate the solution in this equation. Using compensated compactness
arguments the convergence of the approximate solution to the unique entropy solution has been proven.
A well known approach is to use Fourier/Chebyshev or Legendre polynomials for the approximation of
the solution. For these cases the basic questions of accuracy and stability have been discussed for scalar
conservation laws.
In this talk we will focus on a different approximation using Appell polynomials. That way we can
build different kinds of exponential filter relative to the differential operator of the Appell polynomials.
Futhermore we discuss the problem using Appell polynomials, because in general they are not orthogonal
to each other in the same degree. So we have to use the orthonormalization process. We utilize them
(direct or after the orthonormalization process) in the SSV-method. Finally we look out on numerical test
for different Appell polynomials.
** This is a joint work with: Th. Sonar.
C. Pestano-Gabino, La Laguna University, Spain
A Particular Type of Matrix Padé Approximants Inspired by Multivariate
Time Series Models
Abstract
In this paper we define a type of Matrix Padé Approximants inspired by the identification stage of multivariate time series models considering Scalar Component Models. Of course, formalization of certain
properties in the Matrix Padé Approximation framework can be applied to time series models and other
different fields. Specifically, we want to help study Matrix Padé Approximants in the following way: To find
rational representations (or rational approximations) of a matrix formal power series, with matrix numerator
and denominator, and verifying three conditions: a) minimal row degrees for numerator and denominator,
b) invertible denominator in zero, c) canonical representation (without free parameters).
** This is a joint work with: C. González-Concepción and M. C. Gil-Fariña.
M. Derevyagin, TU Berlin, Germany
Darboux transformations of Jacobi matrices and Pade approximation
Abstract
62
CHAPTER 6. THURSDAY, SEPTEMBER 1
Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We
construct a pair of the lower and upper triangular block matrices L and U such that J = LU and the
matrix JC = UL is a monic generalized Jacobi matrix associated with the function FC (λ ) = λ F(λ ) + 1. It
turns out that the Christoffel transformation JC of a bounded monic Jacobi matrix J can be unbounded.
This phenomenon is shown to be related to the effect of accumulating at ∞ of the poles of the Padé
approximants of the function FC although FC is holomorphic at ∞.
The case of the UL-factorization of J is considered as well.
** This is a joint work with: V. Derkach.
6.4 Posters
1. Cristian Arteaga, Departamento de Análisis Matemático Universidad de La Laguna, Spain.
A scheme for interpolation by Hankel translates of a basis function.
Abstract: Golomb and Weinberger (1958) described a variational approach to interpolation which
reduced the problem to minimizing a norm in a reproducing kernel Hilbert space generated by means
of a small number of data points. Later, Duchon (1977) defined radial basis function interpolants as
functions which minimize a suitable seminorm given by a weight in spaces of functions closely related
to Sobolev spaces. These minimal interpolants could be written as a linear combination of translates
of a single function φ , known as a basis function, plus a polynomial. Light and Wayne (1999)
extended Duchon’s class of weight functions which in turn allowed for non-radial basis functions in
their scheme. Following the approach of Light and Wayne, we discuss interpolation of complex-valued
functions defined on the positive real axis I by certain special subspaces. The set of interpolation
points will be a subset {a1 , . . . , an } of I and the interpolants will take the form
n
m−1
u(x) = ∑ αi (τai φ ) (x) +
i=1
∑ β j pµ, j (x)
(x ∈ I),
j=0
where µ > −1/2, φ is the basis function, pµ, j (x) = x2 j+µ+1/2 ( j ∈ Z+ , 0 ≤ j ≤ m − 1) is a Müntz
monomial, τz (z ∈ I) denotes the Hankel translation operator of order µ, and αi , β j (i, j ∈ Z+ , 1 ≤
i ≤ n, 0 ≤ j ≤ m − 1) are complex coefficients. An estimate for the pointwise error of the interpolants
is given.
**This is a joint work with: I. Marrero.
2. F. Lledó, Universidad Carlos III de Madrid, Spain.
Følner sequences in Operator Theory and Operator Algebras.
Abstract In the first part of this talk I will recall the notion and first properties of Følner sequences in
the context of Operator Theory and Operator Algebras. Let T ⊂ L (H ) be a set of bounded linear
operator acting on a complex separable Hilbert space H . An increasing sequence of non-zero finite
rank orthogonal projections {Pn }n∈N strongly converging to IH is called a Følner sequence for T , if
lim
n
kT Pn − Pn T k2
=0,
kPn k2
T ∈T ,
where k · k2 is the Hilbert-Schmidt norm. Følner sequences generalize the notion quasi-diagonality
for operators and can also be applied in spectral approximation problems.
6.4. POSTERS
63
In the second part of the talk I will present recent results in separate joint works with Pere Ara
(U.A.B.) and Dmitry Yakubovich (U.A.M.): I will mention several classes of operators with and
without Følner sequences. E.g. any essentially normal operator has a Følner sequence. Finally, we
will construct Følner sequences for crossed product of von Neumann algebras and give an intrinsic
characterization of this notion in terms of completely positive maps.
3. Héctor Luna-Garcı́a , Universidad Autónoma Metropolitana-Unidad Azcapotzalco, México.
Cálculo de series infinitas mediante algoritmos de funciones poligammas y transformada de Laplace.
Abstract: Se introducen algoritmos mediante las funciones poligammas para obtener lı́mites exactos
de una amplia clase de series infinitas. Además, se introduce la transformada de Laplace con el
mismo propósito y para encontrar la suma de series infinitas convergentes.
Las series infinitas aparecen en la mayorı́a de los campos de la Fı́sica como es el caso de las se∞
ries: ∑
n=1
∞
∑
n=1
1
(n+a)(n+b)
que aparece en la teorı́a de campos cuantizados a temperatura finita1 y la serie
1
(2n−1)(2n+1)(2n+3)
en la determinación del recorrido medio de los neutrones generados en condi-
ciones de isotropı́a. Por esto las series infinitas constituyen una parte importante de un curso fun∞
damental de análisis; la serie ∑
n=1
(−1)n
(2n+1)
es la definición de Leibnitz del irracional π. Para tratar a
las series infinitas es costumbre en la literatura clásica introducir los criterios de convergencia de
D’Alambert, Cauchy, Raabe, Kummer y Gauss2 , que aseguran la convergencia o divergencia de la
serie infinita. Sin embargo, la aplicación de los criterios de convergencia de una serie no proporciona
el lı́mite de ella. En el presente trabajo, se presentan algoritmos que permiten mediante las funciones
eulerianas, calcular con exactitud el lı́mite de una serie convergente y que se aplican a una amplia
clase de series infinitas. El trabajo se divide en cuatro partes: en la sección 1, se ilustra el método
para casos especiales; en la sección 2, se generaliza el método; en la sección 3 se utiliza la transformada de Laplace para la suma de series infinitas y finalmente en la sección 4 se presentan conclusiones.
1. Ramond P., Field Theory: A Modern Primer, 2nd Edition, Sarat Book House, 2007.
2. Arfken G., Mathematical Methods for Physicists, Sixth Edition: A Comprehensive Guide. Elsevier
Academic Press, 2005.
**This is a joint work with: Luz Marı́a Garcı́a-Cruz.
4. J. J. Moreno-Balcázar, University of Almerı́a, Spain.
Jacobi-Sobolev orthogonal polynomials: Asymptotics and Fourier expansions.
Abstract: We obtain asymptotic properties of the sequence of Sobolev orthogonal polynomials with
respect to
N
( f , g)S =
∑
Z 1
f ( j) (x)g( j) (x)dµ,
j=0 −1
where N ≥ 1, and the measure µ corresponds to the Jacobi weight. We also provide a Cohen type
inequality for the corresponding Fourier expansions.
**This is a joint work with: B. Xh. Fejzullahu and F. Marcellán.
64
CHAPTER 6. THURSDAY, SEPTEMBER 1
5. C. Suárez, Universidad de Vigo, Spain.
On second kind polynomials associated with rational transformations of linear functionals.
Abstract: In this paper the following construction process of orthogonal polynomials on the unit circle
is considered:
Let u be a regular and hermitian linear functional and let {Φn } be the corresponding orthogonal
polynomials sequence. We define a new linear functional L by means the following relation with u
λ (z − β )L = (z − α)u, α, β , λ ∈ C, λ 6= 0.
In this situation we obtain conditions for the regularity of L, as well as the corresponding orthogonal
polynomials sequence. Also, we give one explicit representation for the orthogonal polynomials
sequence of the second kind associated to L.
For the particular case when α = β , L becomes in the well-known modification of u by addition of
a Dirac mass. This case will be studied with special attention. Keywords: orthogonal polynomials, unit
circle, Bernstein-Szegö polynomials, Carathéodory function, measure modification.
AMS subject Classification: 42C05
6. T. Nagao, Nagoya University, Japan.
Classical skew orthogonal polynomials related to random matrices.
Abstract: With respect to antisymmetric inner products of polynomials, one can construct skew orthogonal
polynomials by using Gram-Schmidt orthogonalization procedure. Skew orthogonal polynomials are known to
be useful in the analysis of eigenvalue correlations of real and quaternion random matrices.
Classical orthogonal polynomials on continuous measures (Jacobi, Laguerre and Hermite polynomials) and
some discrete measures are accompanied by the corresponding skew orthogonal polynomials. Such ”classical
skew orthogonal polynomials” have particularly useful properties such as compact expansion formulas and
factorizable normalization constants. In this presentation, classical skew orthogonal polynomials are listed and
their characterization is discussed.
Reference: T. Nagao, Pfaffian expressions for random matrix correlation functions, J. Stat. Phys. 129 (2007)
1137.
7. F. Ndayiragije, Katholieke Universiteit Leuven, Belgium.
Multiple Charlier polynomials:generating function,recurrence relation,ratio asymptotics.
Abstract: We investigate multiple Charlier polynomials and we will use his explicit formula to obtain a generating
function which is a generalization of the familiar generating function (the case r = 1) for Charlier polynomials.
We will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two
multiple Charlier polynomials.
**This is a joint work with: W. Van Assche.
8. D. Pérez, Universidad Centroccidental Lisandro Alvarado, Venezuela.
On Markov-Bernstein type inequalities and certain class of Sobolev polynomials.
Abstract: Let (µ0 , µ1 ) be a vector of non-negative measures on the real line, with µ0 not identically zero, finite
moments of all orders, compact or non compact supports, and at least one of them having an infinite number
of points in its support. We show that for any linear operator T on the space of polynomials with complex
coefficients and any integer n ≥ 0, there is a constant γn (T ) ≥ 0, such that
kT pkS ≤ γn (T )kpkS ,
6.4. POSTERS
65
for any polynomial p of degree ≤ n, where γn (T ) is independent of p, and
kpkS =
Z
2
|p(x)| dµ0 (x) +
Z
1
2
|p (x)| dµ1 (x) .
0
2
We find a formula for the best possible value n (T ) of γn (T ) and inequalities for n (T ). Also, we give some
examples when T is a differentiation operator and (µ0 , µ1 ) is a vector of orthogonalizing measures for classical
orthogonal polynomials.
** This is a joint work with: Y. Quintana.
9. Vanessa Avansini Botta Pirani, Universidade Estadual Paulista, Brazil.
On the Radau methods based on Legendre polynomials.
Abstract: The theory of differential equations is part of one area of Mathematics very rich in applications. The
numerical methods for the solution of ordinary differential equations are, in the same way as the equations
themselves, important sources of problems to be studied. As prominence one has the Radau methods which
are important for the solution of stiff problems, that has been very used in the study of natural phenomenas
related to differents areas of the knowledgement. In this paper we use the Legendre polynomials to obtain the
coefficients of some Radau methods and the order stars to analyse some stability questions.
**This is a joint work with: Messias Menguette Jr.
10. C. González Concepción, University of La Laguna, Spain.
Illustration on the relationship between Scalar Component Models and Matrix Padé Approximation.
Abstract: This poster illustrates the feedback on the relationship between Scalar Component Models in Multivariante Time Series Analysis and Matrix Padé Approximation.
First, using properties of Matrix Padé Approximation we have solved an open theoretical question in the initial
identification of Scalar Component Models posed initially by Tiao and Tsay. Next, the ideas used for it in time
series models inspire a new definition of Matrix Padé Approximants.
**This is a joint work with: C. Pestano Gabino and M. C. Gil Fariña.
7
Friday, September 2
7.1 Schedule of the day
Chair of plenary session
09:30 to 10:20
10:30 to 11:20
11:20 to 12:00
Room
Chair of session
12:00 to 12:25
12:30 to 12:55
13:00 to 13:25
Chair of open problem session
13:25 to 15:00
(∗)
1.2.3.4.5.-
Antonio Duran
Plenary lecture by Maria Jose Cantero
Plenary lecture by Alberto Grunbaum
A (Auditorium)
Wolfgang Gawronski
D. Gómez-Ullate
Bernardo de la Calle
Nikos Stylianopoulus
Poster session IV(∗) and coffee break
B (4.1.E01)
C (4.1.E03)
José L. López
Alejandro Zarzo
M. van Pruijssen
Corina Grosu
Pablo Román
Johann Brauchart
Howard Cohl
Ana Folquie
Bernhard Beckermann
Lunch + Open Problems Session
The following posters will be presented
Ana Portilla.
6.Fernando Rodrigo Rafaeli.
7.Daniel Alberto Rivero Castillo.
8.José Manuel Rodrı́guez Garcı́a. 9.Luz Roncal.
10.-
D (4.1.E04)
Franck Wielonsky
Ionela Moale
Maxim Yattselev
Maryna Viazovska
during the Poster Session of the day.
Anier Soria-Lorente.
Juan Luis Varona.
Nelson Vieira.
Alejandro Zarzo Altarejos.
Swaminathan Anbhu.
7.2 Plenary lectures
Maria José Cantero, Universidad de Zaragoza, Spain.
Title: Spectral transformations of hermitian linear functionals.
Abstract
In this talk we present some recent results concerning rational modifications of moment functionals on the
unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to
study arbitrary rational modifications. The main objective is the characterization of the quasi-definiteness
of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur
parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian
functionals, so that the maximum number of orthogonal polynomials is characterized by the number of
consistent steps of an algorithm based on the referred recurrence for the Schur parameters.
67
68
CHAPTER 7. FRIDAY, SEPTEMBER 2
The effectiveness of this new approach is shown in several examples which also lead to the discovery
of new families of orthogonal polynomials on the unit circle, some of them with respect to non positive
definite functionals.
Paper joint with L. Moral and L. Velázquez
Alberto Grunbaum, University of Berkeley, USA.
Title: Quantum walks and CMV matrices.
Abstract
The study of certain classical Markov chains by spectral methods is a classical area going back at least
to work by S.Karlin and J. McGregor. Recently this approach has been undertaken in the case of certain
Quantum Walks. Of particular interest is the relation between the orthogonality measure and the limit law
for the distribution of the displacement of the quantum walker, as well as issues such as recurrence and
localization. We will discuss some general results as well as some specific examples.
7.3 Short communications
7.3.1 Room A (Auditorium)
D. Gómez-Ullate, Universidad Complutense de Madrid, Spain
Two-step Darboux transformations and exceptional Laguerre polynomials
Abstract
It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist,
that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we
show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step
algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux
transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial
system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre
and the type II X` -Laguerre polynomials are recovered.
** This is a joint work with: N. Kamran and R. Milson.
B. de la Calle Ysern, Polytechnic University of Madrid, Spain
Optimal extension of the Szego quadrature
Abstract
All the possible extensions of the Szegő quadrature with the highest degree of exactness and self-reciprocal
nodal polynomial are constructed. For a large class of weight functions we prove that the nodal polynomial
7.3. SHORT COMMUNICATIONS
69
fulfills strong asymptotic behavior uniformly on the unit circle. This asymptotic representation is analogous
to that of para-orthogonal polynomials. We additionally prove that, for the class of weight functions
considered and sufficiently large number of nodes, the extended quadratures have positive weights and
simple nodes on the unit circle. Finally, some interesting consequences related to Gauss-Kronrod quadrature
in the real line are discussed.
N. Stylianopoulos, University of Cyprus, Cyprus
Bergman Orthogonal Polynomials on Archipelaga: Construction,
Asymptotics, Short Recurrences and Shape Recovery
Abstract
Let G := ∪Nj=1 G j be the union of N mutually exterior, bounded domains in the complex plane and let
{pn }∞
n=0 denote the sequence of Bergman polynomials of G. This is defined as the sequence
pn (z) = λn zn + · · · ,
λn > 0,
n = 0, 1, 2, . . . ,
R
of polynomials that are orthonormal with respect to the inner product h f , gi := G f (z)g(z)dA(z), where dA
stands for the area measure. (In the case when N > 1, we call G an archipelago.)
The purpose of the talk is to present some recent developments regarding the construction, the theory
and the applications of Bergman polynomials. These developments include:
(i) A stable Arnoldi Gram-Schmidt process for constructing orthonormal polynomials.
(ii) Ratio asymptotics and finite-term recurrence relations for Bergman polynomials.
(iii) An reconstruction algorithm for recovering the shape of G in the single component case N = 1 from
a finite section of its complex moment matrix,
Z
µi j :=
zi z j dA(z), 0 ≤ i, j ≤ n,
G
using ratio asymptotics. The importance of the algorithm in (iii) is underlined by the fact that suitable
tomographic data, for example, parallel ray measurements in 2D geometric tomography, can be transformed
into a finite section of the moment matrix.
7.3.2 Room B (4.1.E01)
M. van Pruijssen, Radboud Universiteit, FNWI, IMAPP, Wiskunde, Netherlands
Matrix Valued Orthogonal Polynomials Related to a Compact Symmetric
Pair of Rank One
Abstract
70
CHAPTER 7. FRIDAY, SEPTEMBER 2
The connection between special functions and the representation theory for compact Lie groups is a very
fruitful one. In the compact rank one case U/K one can construct the Jacobi polynomials from the Kbi-invariant functions on G. This group theoretic point of view gives explicitly the weight with respect to
which the Jacobi polynomials are orthogonal. Moreover, one obtains explicitly a second order differential
operator and recurrence relations for the family of Jacobi polynomials in terms of data from the group.
In this talk I discuss the construction of a family of matrix valued orthogonal poynomials (MVOP) for
some specific compact symmetric pairs (U, K) of rank one. The construction indicates how to calculate
explicitly (i) a weight with respect to which the family is orthogonal (ii) a three term recurrence relation
for the family and (iii) a symmetric differential operator with the polynomials as eigenfunctions.
We present the families of MVOPs that arise in this fashion from the pair (SU(2) × SU(2), diag). The
families of MVOPs that we obtain are essentially different from the families of MVOPs associated to the
pair (SU(3),U(2)) that have been calculated by Grünbaum et al. Moreover, an interesting symmetry from
the group is present on the level of the MVOPs.
** This is a joint work with: E. Koelink and P. Román.
P. M. Román, Katholieke Universiteit Leuven, Belgium
Harmonic analysis in the complex hyperbolic plane and the matrix valued
hypergeometric function
Abstract
For a locally compact group G with a compact subgroup K, the spherical transform on the convolution
algebra Cc,δ (G) and its corresponding inversion formula are discussed. Here Cc,δ (G) denotes the the algebra
of all continuous functions f with compact support on G such that χ δ ∗ f = f ∗ χ δ = f and χδ denotes the
character of a unitary irreducible representation of K times its dimension.
The case of the pair G = SU(2, 1) and K = U(2) is considered in full generality, i.e. for any K-type.
Since we work with spherical functions of an arbitrary K-type, the spherical functions are now described
explicitly in terms of matrix valued hypergeometric functions. In this way we obtain expressions for the
spherical transform and the corresponding inversion formula which involve matrix hypergeometric functions
of arbitrary size. We obtain an inversion formula for the spherical transform by using the Fourier inversion
formula in G. If we consider K-types of dimension one the spherical transform reduces to a multiple of the
Jacobi transform.
** This is a joint work with: J. Tirao.
H. S. Cohl, National Institue of Standards and Technology, USA
Fourier expansions for logarithmic fundamental solutions of the
polyharmonic equation
Abstract
In even-dimensional Euclidean space, for integer powers of the Laplacian greater than or equal to the
dimension divided by two, a fundamental solution of the polyharmonic equation has logarithmic behavior.
7.3. SHORT COMMUNICATIONS
71
We give two approaches for developing an azimuthal Fourier expansion for this fundamental solution. The
first approach relies upon the solution of a recurrence relation for a certain set of polynomials. We compute
a generating function for these polynomials and derive an azimuthal Fourier expansion for a fundamental
solution of the polyharmonic equation in terms of these polynomials. The second approach uses parameter
derivatives applied to a generalized Heine’s identity for complex Fourier series of binomials. The Fourier
coeficients of Heine’s identity are given in terms of associated Legendre functions of the second kind with
odd-half-integer degree and real argument greater than unity. Through Whipple’s formulae for associated
Legendre functions, we take advantage of a parameter derivative result by Szmytkowski (for associated
Legendre functions of the first kind with integer-degree and integer-order) to obtain an azimuthal Fourier
expansion for a fundamental solution the polyharmonic equation. This expansion is given in terms of sums
over associated Legendre functions. We conclude by comparing the two approaches.
7.3.3 Room C (4.1.E03)
Corina Grosu, University POLITEHNICA of Bucharest, Romania
Hamiltonians on time scale structures
Abstract
Let T be a time scale (a closed nonempty subset of R ). Consider a Hamiltonian function H , where
H : T × Mn → R such that the following conditions are satisfied:
1. H (t, x) is rd-continuous for every t ∈ T
2. the partial derivatives of first and second order of H (t, x) relative to x ∈ Mn exist and are continuous
3. Mn is a proper affine sphere and T × Mn is a warped product endowed with a semi-Riemannian metric
containing the perturbation factor f (t),
4. the perturbation factor f : T → R+ , f (t) > 0 for every t ∈ T, is completely delta differentiable in
every t ∈ T
5. the potential function V (t, x)is associated to a sequence of time dependent orthogonal polynomials
connected to Jacobi’s elliptic functions
By using the factorization method in curvilinear coordinates along with existing results concerning the
classification of affine hyperspheres, we study the transformation of the sequence of orthogonal polynomials
when performing Laplace-Darboux transformations on the manifold Mn .
J. S. Brauchart, University of New South Wales, Australia
Weighted minimal Riesz energy points respecting rotational symmetry
Abstract
72
CHAPTER 7. FRIDAY, SEPTEMBER 2
Let A be a compact set in the right-half of the complex plane and Γ(A) the set of revolution obtained by
rotating the complex plane about the imaginary axis. The discrete weighted s-energy associated with a
weight function w(x, y) respecting the rotational symmetry of Γ(A) seeks to find N-point configurations
minimizing the weighted s-energy functional
∑ w(x j , xk ) ks (x j , xk )
(e.g. w(x, y) = [ρ(x)ρ(y)]−s/4 )
j6=k
among all N-point configurations in Γ(A). The kernel function ks is the logarithmic kernel log(1/|x − y|) or
the Riesz s-kernel 1/|x − y|s (s > 0). By rotational symmetry, the problem on Γ(A) in R3 for the singular
kernel function ks can be reduced to a weighted energy problem on A in C for some new kernel Ks . We
present results for the planar weighted energy problem for the reduced kernel Ks and the implications for
the 3-dimensional problem.
** This is a joint work with: D. Hardin, E. Saff (Vanderbilt University) and R. Womersley (UNSW).
A. Foulquié, Universidade de Aveiro, Portugal
Integrable systems, multiple orthogonal polynomials and operator theory
Abstract
Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based
on the analysis of the matrical moments for the main operator of the pair. The solutions of these systems
are studied in terms of properties of this operator, giving, under some conditions, explicit expressions for
the resolvent function. We also relate the solutions of two given discrete full Kostant Toda lattice by means
of a Darboux type transformation.
** This is a joint work with: D. Barrios Rolanı́a and A. Branquinho.
7.3.4 Room D (4.1.E04)
I. Moale, Johannes Kepler University Linz, Australia
On complex (non-analytic) Chebyshev polynomials in C2
Abstract
We consider the problem of finding a best uniform approximation to the standard non-analytic monomial
on the unit ball in C2 by polynomials of lower degree with complex coefficients. We reduce the problem to
a one-dimensional weighted approximation problem on an interval. In a sense, the corresponding extremal
polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on so-called comb-like domains. In these terms, the value
of the minimum deviation and the representation for a polynomial of best approximation for the original
problem are given. Furthermore, we derive asymptotics for the minimum deviation.
** This is a joint work with: P. Yuditskii.
7.4. POSTERS
73
M. Yattselev, University of Oregon, USA
Asymptotics of Nuttall-Stahl polynomials
Abstract
We consider asymptotic behavior of Pade approximants to Cauchy integrals of Jacobi-type densities defined
on contours of minimal capacity.
** This is a joint work with: A. I. Aptekarev.
M. Viazovska,, Max Planck Institute for Mathematics, Germany
On optimal asymptotic bounds for spherical t- designs
Abstract
A spherical t-design is a finite set of N points on the d-dimensional unit sphere Sd such that the average
value of any polynomial P of degree t or less on this set is equal to the average value of P on the whole
sphere. For each N ≥ cd t d we prove the existence of a spherical t-design on the sphere Sd consisting of
N points, where cd is a constant depending only on d. This result proves the conjecture of Korevaar and
Meyers concerning an optimal order of minimal number of points in a spherical t-design on Sd for a fixed
d.
** This is a joint work with: A. Bondarenko and D. Radchenko.
7.4 Posters
1. Ana Portilla, St. Louis University, Spain.
The multiplication operator, zero location and asymptotic for non-diagonal Sobolev norms.
Abstract: In this paper we are going to study the zero location and asymptotic behavior of extremal
polynomials with respect to a generalized non-diagonal Sobolev norm in which the product of the
function and its derivative appears. The orthogonal polynomials with respect to this Sobolev norm
are a particular case of those extremal polynomials. The multiplication operator by the independent
variable is the main tool in order to obtain our results.
** This is a joint work with: J. M. Rodrı́guez and E. Tourı́s.
2. Fernando Rodrigo Rafaeli, Universidade Estadual Paulista, Brazil.
Monotonicity of Zeros Orthogonal Polynomials.
Abstract: Let {pn (x)}∞
n=0 be a sequence of orthogonal polynomials with respect to a positive Borel
measure µ(x) supported in the interval (ξ , η). For every a 6∈ (ξ , η) and τ ∈ I ⊂ R, we study the zeros
xn,k (τ, a) of the polynomial pn (τ, a; x) which are orthogonal with respect to the measure
dµτ (a; x) = (x − a)τ dµ(x).
(7.1)
74
CHAPTER 7. FRIDAY, SEPTEMBER 2
3. Daniel Alberto Rivero Rodrı́guez, Universidad Carlos III de Madrid, Spain.
Computing Sobolev orthogonal polynomials.
Abstract: In this work we propose a new algorithm for generating Sobolev orthogonal polynomials
and present numerical examples to compare with the moment-based methods, known as “Modified
Chebyshev algorithm” and “Stieltjes’s algorithm”.
** This is a joint work with: H. Pijeira.
4. Jose M. Rodrı́guez, Universidad Carlos III de Madrid, Spain.
Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms.
Abstract: In this poster we are going to study the zero location and asymptotic behavior of extremal
polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic
form is allowed to degenerate. The orthogonal polynomials with respect to this Sobolev norm are
a particular case of those extremal polynomials. The multiplication operator by the independent
variable is the main tool in order to obtain our results.
** This is a joint work with: A. Portilla, Y. Quintana and E. Tourı́s.
5. L. Roncal, Universidad de La Rioja, Spain.
Wave equation for the Bessel Laplacian
Abstract We study some inequalities concerning the solutions to the Cauchy problem associated to
the wave equation in the multidimensional ball Bd , d ≥ 1. We will consider the radial case, hence
the Cauchy problem is the following
( 2
∂
u(t, x) = Lν u(t, x),
∂t 2
∂
u(0, x) = ψ(x).
u(0, x) = φ (x), ∂t
where
Lν = −∆ −
2ν + 1 d
,
x dx
with the type index ν = d/2 − 1.
** This is a joint work with: O. Ciaurri.
6. A. Soria-Lorente, Universidad Carlos III de Madrid, Spain.
On some second order linear difference equations leading to the irrationality of ζ (3).
Abstract: Other sequences of rational approximants -different from the Apéry’s one- leading to the
irrationality of ζ (3) are given.
** This is a joint work with: J. Arvesú.
7. C. Suárez, Universidad de Vigo, Spain.
On second kind polynomials associated with rational transformations of linear functionals.
Abstract: In this paper the following construction process of orthogonal polynomials on the unit circle
7.4. POSTERS
75
is considered:
Let u be a regular and hermitian linear functional and let {Φn } be the corresponding orthogonal
polynomials sequence. We define a new linear functional L by means the following relation with u
λ (z − β )L = (z − α)u, α, β , λ ∈ C, λ 6= 0.
In this situation we obtain conditions for the regularity of L, as well as the corresponding orthogonal
polynomials sequence. Also, we give one explicit representation for the orthogonal polynomials
sequence of the second kind associated to L.
For the particular case when α = β , L becomes in the well-known modification of u by addition of
a Dirac mass. This case will be studied with special attention. Keywords: orthogonal polynomials, unit
circle, Bernstein-Szegö polynomials, Carathéodory function, measure modification.
AMS subject Classification: 42C05
8. Juan L. Varona, Universidad de La Rioja, Spain.
Asymptotic behavior of the Lerch transcendent function.
Abstract: In our recent paper “Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials” (to
appear in Math. Comp.), we found asymptotic estimates for the Apostol-Bernoulli polynomials Bn (z; λ ) when
n → ∞, uniformly on compact subsets of the complex plane. We do it by mean of a simple procedure that
does not require complex variable theory. This procedure has three steps:
a) Firstly, to get the Fourier series of Bn (x; λ ) on the interval [0, 1]. From this series it is easy to find
uniform asymptotic estimates on [0, 1].
b) Secondly, to find a formula that allows to relate the Bernoulli polynomials on z ∈ C with the value at a
fixed point on [0, 1]. For this purpose, the well known (and easy to proof) “umbral property”
n n
Bn (z; λ ) = ∑
Bn (0; λ ) zk .
k
k=0
has been enough.
c) The last step requires to use this formula to translate the asymptotic estimates of the Fourier series on
[0, 1] to the whole complex plane.
By mean of
Bn (z; λ )
,
n
the Apostol-Bernoulli polynomials are a particular case of the Lerch transcendent function
Φ(λ , 1 − n, z) = −
∞
Φ(λ , s, z) =
λk
∑ (k + z)s .
k=0
We will see that the above mentioned procedure can be adapted to get asymptotic estimates for this function.
**This is a joint work with: L. M. Navas and F. J. Ruiz.
9. N. Vieira, Center of Mathematics, University of Porto, Portugal.
Index Transform in an analysis of the time dependent Schrödinger equation
Abstract In this talk we introduce a notion of Schrödinger’s kernel to the familiar Kontorovich-Lebedev transform. In order to control its singularity at infinity, we will need to implement the so-called regularization
procedure. Hence we will obtain a sequence of regularized kernels which converge to the original kernel when a
regularization parameter tends to zero. We study differential and semigroup properties of the regularized kernel
76
CHAPTER 7. FRIDAY, SEPTEMBER 2
and construct fundamental solutions of a regularized time-dependent Schrödinger’s equation. The correspondent regularized integral transformation is presented. We also establish analogs of the classical Heisenberg
inequality and uncertainty principle for this transformation. Finally, we examine a pointwise convergence of
this family of regularized integral operators, when the regularization parameter tends to zero.
** This is a joint work with: S. Yakubovich (University of Porto).
10. A. Zarzo Altarejos, Universidad Politécnica de Madrid, Spain.
Kravchuk polynomials in two discrete variables: Inversion and connection problems and the bivariate NAVIMA
algorithm
Abstract Let x = (x, y) ∈ R2 , and let xn (n ∈ N0 ) be the column vector of the products xn−k yk arranged in the
graded lexicographical order, i.e.:
xn = xn , xn−1 y, . . . , xn−k yk , . . . , xyn−1 , yn .
Let K̂np11,n,p22 (x, y; N) be the bivariate monic Kravchuk polynomials of total degree n = n1 + n2 and denote by
Knp1 ,p2 (x, y; N) the column polynomial vector:
p1 ,p2
p1 ,p2
p1 ,p2
Knp1 ,p2 (x, y; N) = (K̂n,0
(x, y; N), . . . , K̂n−k,k
(x, y; N), . . . , K̂0,n
(x, y; N))T .
The inversion problem for this vector polynomial family asks for the computation of the matrix coefficients
Im (n) of size (n + 1) × (m + 1) in the expansion:
n
xn =
∑ Im (n)Knp1 ,p2 (x, y; N)
m=0
Our aim here is to give a recurrent procedure to solve this problem. This is done using the structural properties
satisfied by these polynomials [1] which allow us to develop an extension to this bi-variate situation of the
phNAVIMA algorithm [2], developed by the authors for one variable problems. As a byproduct, the solution
of a connection problem between two bi-orthogonal bivariate Kravchuk families is given. [1] J. Rodal, I. Area
and E. Godoy. J. Math. Anal. Appl. 340 (2008) 825–844. [2] I. Area, E. Godoy, A. Ronveaux and A. Zarzo.
J. Comput. Appl. Math. 89 (1998) 309–325.
** This is a joint work with: I. Area, E. Godoy, J. Rodal and A. Ronveaux.
11. Swaminathan Anbhu, Indian Institute of Technology Roorkee, India.
Pick functions and Chain Sequences for hypergeometric type functions
Abstract The main objective of this work is to find the members of the class of Pick functions studied
by Nevanlinna using moment problems. These members are identified using chain sequences appearing as
Schur-Wall-g-fraction representation of certain analytic self maps of the unit disc |z|.
8
Committees
8.1 Scientific Committee.
• Aleksander Aptekarev, Keldysh Institute of Applied Mathematics, Russia.
• Bernhard Beckermann, Université des Sciences et Technologies de Lille, France.
• Christian Berg, University of Copenhagen, Denmark.
• Antonio Durán Guardeño, Universidad de Sevilla, Spain.
• Jeff Geronimo, Georgia Institute of Technology, USA.
• Arno Kuijlaars, Katholieke Universiteit Leuven, Belgium.
• Guillermo López Lagomasino, Universidad Carlos III de Madrid, Spain.
• Andrei Martı́nez Finkelshtein, Universidad de Almerı́a, Spain.
• Paul Nevai, Ohio State University, USA.
• Jesús Sánchez Dehesa, Universidad de Granada, Spain.
• Herbert Stahl, Technische Fachhochschule Berlin, Gernamy.
8.2 Organizing Committee.
Chairman. Guillermo López Lagomasino, Universidad Carlos III de Madrid, Spain.
Secretary. Jorge Arvesú Carballo, Universidad Carlos III de Madrid, Spain.
Members.
• Renato Álvarez Nodarse, Universidad de Sevilla, Spain.
• Dolores Barrios Rolanı́a, Universidad Politécnica de Madrid, Spain.
• Héctor Pijeira Cabrera, Universidad Carlos III de Madrid, Spain.
• Mariza Rezola, Universidad de Zaragoza, Spain.
• Alejandro Zarzo Altarejos, Universidad Politécnica de Madrid, Spain.
77
78
CHAPTER 8. COMMITTEES
8.3 Local Committee.
Chairman. Guillermo López Lagomasino, Universidad Carlos III de Madrid, Spain.
Secretary. Jorge Arvesú Carballo, Universidad Carlos III de Madrid, Spain.
Members.
• Alfredo Deaño Cabrera, Universidad Carlos III de Madrid, Spain.
• Fernando Lledó Macau, Universidad Carlos III de Madrid, Spain.
• Héctor Pijeira Cabrera, Universidad Carlos III de Madrid, Spain.
• Eva Touris Lojo, Universidad Autónoma de Madrid, Spain.
• Jorge A. Borrego Morel, Universidad Carlos III de Madrid, Spain.
• Kenier Castillo Rodrı́guez, Universidad Carlos III de Madrid, Spain.
• Edmundo J. Huertas Cejudo, Universidad Carlos III de Madrid, Spain.
• Sergio Medina Peralta, Universidad Carlos III de Madrid, Spain.
• Ma. Francisca Pérez Valero, Universidad Carlos III de Madrid, Spain.
• Daniel A. Rivero Castillo, Universidad Carlos III de Madrid, Spain.
• Anier Soria Lorente, Universidad Carlos III de Madrid, Spain.
9
List of Participants by alphabetical order
1. Alfaro, Manuel. Universidad de Zaragoza. Spain.
alfaro@unizar.es
2. Alfaro, Marı́a Pilar. Universidad de Zaragoza. Spain.
palfaro@unizar.es
3. Álvarez-Estrada, Ramon F. Universidad Complutense. Spain.
rfa@fis.ucm.es
4. Álvarez Fernández, Carlos. Universidad Pontificia Comillas. Spain.
calvarez@cee.upcomillas.es, albireo1976@gmail.com
5. Álvarez Nodarse, Renato. Universidad de Sevilla. Spain.
ran@us.es
6. Anbhu, Swaminathan. Department of Mathematics, I.I.T. Roorkee. India.
swamifma@iitr.ernet.in, mathswami@gmail.com
7. Aptekarev, Alexander. Keldysh Institute of Applied Mathematics. Russia.
aptekaa@spp.keldysh.ru
8. Area, Iván. Universidad de Vigo. Spain.
area@dma.uvigo.es
9. Arteaga, Cristian. La Laguna University. Spain
cclement@ull.es
10. Arvesú Carballo, Jorge. Universidad Carlos III de Madrid. Spain.
jarvesu@math.uc3m.es
11. Atia, Mohamed. Faculté des sciences de Gabes. Tunisia.
jalel.atia@gmail.com, jalel.atia@fsg.rnu.tn
12. Barrios Rolanı́a, Dolores. Universidad Politécnica de Madrid. Spain.
dbarrios@fi.upm.es
13. Batista, David. San Diego State University -LBNL. USA.
numerico@gmail.com
79
80
CHAPTER 9. LIST OF PARTICIPANTS
14. Beckermann, Bernhard. Université des Sciences et Technologies de Lille. France.
bbecker@math.univ-lille1.fr
15. Bello Hernández, Manuel. Universidad de La Rioja. Spain.
mbello@unirioja.es
16. Berdysheva, Elena. University of Hohenheim. Germany.
elena.berdysheva@uni-hohenheim.de
17. Berg, Christian. University of Copenhagen. Denmark.
berg@math.ku.dk
18. Bernard, J. M. L. CEA-DIF. France.
jean-michel.bernard@cea.fr
19. Berriochoa, Elias. Universidad de Vigo. Spain.
esnaola@uvigo.es
20. Betea, Dan. California Institute of Technology. USA.
dan.betea@caltech.edu
21. Borrego Morell, Jorge Alberto. Universidad Carlos III de Madrid. Spain.
jbmorell@gmail.com
22. Bosuwan, Nattapong. Vanderbilt University. USA.
nattapong.bosuwan@vanderbilt.edu, natta_cu@hotmail.com
23. Brahim, Kamel. Faculty of Sciences Tunis. Tunisia.
kamel710@yahoo.fr
24. Branquinho, Amilcar. Universidade de Coimbra. Portugal.
ajplb@mat.uc.pt
25. Brauchart, Johann. University of New South Wales. Australia.
j.brauchart@unsw.edu.au
26. Bultheel, Adhemar. K.U.Leuven, Dept. Computer Science. Belgium.
adhemar.bultheel@cs.kuleuven.be
27. Cachafeiro, Alicia. Universidad de Vigo. Spain.
acachafe@uvigo.es
28. Cacoq, Junot. Universidad Carlos III de Madrid. Spain.
jcacoq@math.uc3m.es
29. Cala Rodrı́guez, Francisco. Universidad Austral de Chile. Chile.
fcala@uach.cl
30. Cantero, Marı́a José. Universidad de Zaragoza. Spain.
mjcante@unizar.es
31. Cardoso, José. Universidade de Trás-os-Montes e Alto Douro. Portugal
jluis@utad.pt
81
32. Cassatella, Giovanni A. Universidad Complutense. Spain.
gcassatella@gmail.com
33. Castillo, Kenier. Universidad Carlos III de Madrid. Spain.
kcastill@math.uc3m.es
34. Castro Smirnova, Mirta Marı́a. Universidad de Sevilla. Spain.
mirta@us.es
35. Chaggara, Hamza. Ecole Supérieure des Sciences et de Tech. de Sousse. Tunisia.
hamza.chaggara@ipeim.rnu.tn
36. Chihara, Theodore. Purdue University Calumet. USA.
chihara@purduecal.edu, tschihara@comcast.net
37. Christiansen, Jacob. California Institute of Technology. USA.
stordal@caltech.edu
38. Ciaurri, Óscar. Universidad de La Rioja. Spain.
oscar.ciaurri@unirioja.es
39. Claeys, Tom. Université Catholique de Louvain. Belgium.
tom.claeys@uclouvain.be
40. Clarkson, Peter. University of Kent. United Kingdom.
P.A.Clarkson@kent.ac.uk
41. Cohl, Howard. National Institue of Standards and Technology. USA.
hcohl@nist.gov
42. Connett, William, University of Missouri, USA.
43. Costas Santos, Roberto. College of William and Mary. USA.
rscosa@gmail.com
44. Cotrim, Luı́s. Escola Superior de Tecnologia e Gestão do IPL. Portugal.
luis.cotrim@ipleiria.pt
45. Cruz-Barroso, Ruyman. Katholieke Universiteit Leuven. Spain.
szegoquadrature@hotmail.com
46. De Bie, Hendrik. Ghent University. Belgium.
Hendrik.DeBie@UGent.be
47. De la Calle Ysern, Bernardo. Universidad Politécnica de Madrid. Spain.
bcalle@etsii.upm.es
48. Deaño Cabrera, Alfredo. Universidad Carlos III de Madrid. Spain.
alfredo.deanho@uc3m.es
49. Delgado, Antonia. Universidad de Granada. Spain.
amdelgado@ugr.es
82
CHAPTER 9. LIST OF PARTICIPANTS
50. Delvaux, Stiven. Katholieke Universiteit Leuven. Belgium.
steven.delvaux@wis.kuleuven.be
51. Derevyagin, Maksym. TU Berlin. Germany.
derevyagin.m@gmail.com
52. Deschout, Klaas. Katholieke Universiteit Leuven. Belgium.
klaas.deschout@wis.kuleuven.be
53. Dhaouadi, Lazhar. IPEI Bizerte. Tunisia.
lazhardhaouadi@yahoo.fr
54. Dı́az Mendoza, Carlos. Universidad de La Laguna. Spain.
cjdiaz@ull.es
55. Dobrucky, Branislav. University of Zilina. Slovak Republic.
dobrucky@fel.uniza.sk, branislav.dobrucky@chello.sk
56. Domı́nguez de la Iglesia, Manuel. Universidad de Sevilla. Spain.
mdi29@us.es
57. Dueñas Ruiz, Herbert. Universidad Nacional de Colombia. Colombia.
haduenasr@unal.edu.co
58. Durán Guardeño, Antonio. Universidad de Sevilla. Spain.
duran@us.es
59. Erb, Wolfgang. University of Luebeck. Germany.
erb@math.uni-luebeck.de
60. Escribano, Carmen. Universidad Politécnica de Madrid. Spain.
cescribano@fi.upm.es
61. Fernández Moncada, Paulo Enrique. Universidad Carlos III de Madrid. Spain.
pefernandezm@gmail.com
62. Fernández Morales, Héctor Raúl. Universidad Carlos III de Madrid. Spain.
hectorraulfm@gmail.com, hfernand@math.uc3m.es
63. Ferreira, Chelo. Universidad de Zaragoza. Spain.
cferrei@hotmail.com
64. Fidalgo Prieto, Ulises. Universidad Carlos III de Madrid. Spain.
ulisesfidalgoprieto@yahoo.es
65. Fitouhi, Ahmed. Faculty of Sciences Tunis. Tunisia.
ahmed.fitouhi@fst.rnu.tn, afitouhi@yahoo.fr
66. Foulquié Moreno, Ana. Universidade de Aveiro. Portugal.
foulquie@ua.pt
67. Foupouagnigni, Mama. University of Yaounde I. Cameroon.
foupouagnigni@gmail.com
83
68. Fung, Kei. City University of Hong Kong. China.
charleslkf8571@gmail.com, charleslee8571@hotmail.com
69. Garcı́a Amor, José Manuel. Universidad de Vigo. Spain.
garciaamor@edu.xunta.es
70. Garcı́a Garcı́a, Antonio. Universidad Carlos III de Madrid. Spain.
agarcia@math.uc3m.es
71. Garza Gaona, Luis. Universidad de Colima. Mexico.
garzaleg@gmail.com, luis_garza1@ucol.mx
72. Gascón, José. Universidad Nacional Abierta. Venezuela.
jogascon@una.edu.ve
73. Gawronski, Wolfgang. University of Trier. Germany.
gawron@uni-trier.de
74. Geronimo, Jeff. Georgia Institute of Technology. USA.
geronimo@math.gatech.edu
75. Geudens, Dries. Katholieke Universiteit Leuven. Belgium.
dries.geudens@wis.kuleuven.be
76. Godoy, Eduardo. Universidad de Vigo. Spain.
egodoyo@dma.uvigo.es
77. Gómez-Ullate, David. Universidad Complutense de Madrid. Spain.
david.gomez-ullate@fis.ucm.es, david.gomezullate@gmail.com
78. González, Concepción. La Laguna University. Spain.
cogonzal@ull.es
79. González-Vera, Pablo. Universidad de la Laguna. Spain.
pglez@ull.es
80. Guesmi, Sana. IPEI Bizerte. Tunisia.
guesmisana@yahoo.fr
81. Grandati, Yves. Université Paul Verlaine-Metz. France.
grandati@yahoo.fr, grandati@univ-metz.fr
82. Griffin, James. American University of Sharjah. United Arab Emirates.
jgriffin@aus.edu, JamesCGriffin@gmail.com
83. Grosu, Corina. University POLITEHNICA of Bucharest. Romania.
grosu_c1990@yahoo.com
84. Grunbaum, Alberto. University of Berkeley. USA.
grunbaum@math.berkeley.edu
85. Haneczok, Maciej. Katholieke Universiteit Leuven. Belgium.
maciej.haneczok@wis.kuleuven.be
84
CHAPTER 9. LIST OF PARTICIPANTS
86. Hardy, Adrien. K. U. Leuven. Belgium.
adrien.hardy@wis.kuleuven.be, adrienhardy@yahoo.fr
87. Huertas Cejudo, Edmundo. Universidad Carlos III de Madrid. Spain.
ehuertas@math.uc3m.es, ehuertasce@cofis.es
88. Ibort, Alberto. Universidad Carlos III de Madrid. Spain.
albertoi@math.uc3m.es
89. Iliev, Plamen. Georgia Institute of Technology. USA.
iliev@math.gatech.edu
90. Illán, Jesús. Universidad de Vigo. Spain.
jillan@uvigo.es
91. Jiménez Paiz, Mateo. Universidad de La Laguna. Spain.
mjimenez@ull.es
92. Kalyagin, Valery. Higher School of Economics. Russia.
vkalyagin@hse.ru
93. Kamioka, Shuhei. Kyoto University. Japan.
kamioka@i.kyoto-u.ac.jp
94. Karp, Dymitry. Institute of Applied Mathematics. Russia.
dimkrp@gmail.com
95. Kim, Gwang Hui. KangNam University. South Korea.
ghkim@kangnam.ac.kr
96. Koelink, Erik. Radboud Universiteit. The Netherlands
e.koelink@math.ru.nl
97. Kononova, Anna. Baltic State Technical University. Russia.
anya.kononova@gmail.com
98. Koornwinder, Tom. University of Amsterdam. Netherlands.
T.H.Koornwinder@uva.nl
99. Koumandos, Stamatis. University of Cyprus. Cyprus.
skoumand@ucy.ac.cy
100. Krueger, Helge. Caltech. USA.
helge@caltech.edu
101. Kuijlaars, Arno. Katholieke Universiteit Leuven. Belgium.
arno.kuijlaars@wis.kuleuven.be
102. Kushel, Volha. TU Berlin. Germany.
kushel@math.tu-berlin.de, kushel@mail.ru
103. Lamiri, Imed. Ecole Supérieure des Sciences et Technologie H. Sousse. Tunisia
Imed.Lamiri@infcom.rnu.tn, imed_lamiri@yahoo.fr
85
104. Lee, Jung Rye. Daejin University. South Korea.
charleslkf8571@gmail.com, charleslee8571@hotmail.com
105. Lin, Yu. City University of Hong Kong. China.
linyu1018@gmail.com, linyu2@cityu.edu.hk
106. Lledó Macau, Fernado. Universidad Carlos III de Madrid. Spain.
flledo@math.uc3m.es
107. López, José L. State University of Navarra. Spain.
jl.lopez@unavarra.es
108. López Garcı́a, Abey. Katholieke Universiteit Leuven. Belgium
abey.lopezgarcia@wis.kuleuven.be, abey.lopez@gmail.com
109. López Lagomasino, Guillermo. Universidad Carlos III de Madrid. Spain.
lago@math.uc3m.es
110. Lubinsky, Doron. Georgia Institute of Technology. USA.
lubinsky@math.gatech.edu
111. Lukic, Milivoje. California Institute of Technology. USA.
milivoje.lukic@gmail.com
112. Luna, Héctor. Universidad Autónoma Metropolitana. Mexico.
lghm@correo.azc.uam.mx, 5553972854@prodigy.net.mx
113. Marcellán Español, Francisco. Universidad Carlos III de Madrid. Spain.
pacomarc@ing.uc3m.es
114. Marcokova, Mariana. University of Zilina. Slovak Republic.
mariana.marcokova@fpv.uniza.sk
115. Martı́nez Brey, Eduardo. Universidad de Vigo. Spain.
ebrey@uvigo.es
116. Martı́nez Finkelshtein, Andrei. Universidad de Almerı́a. Spain.
andrei@ual.es
117. Martı́nez González, Pedro. Universidad de Almerı́a. Spain.
pmartine@ual.es
118. Medem, Juan Carlos. Universidad de Sevilla. Spain.
jcmedem@us.es
119. Medina Peralta, Sergio . Universidad Carlos III de Madrid. Spain.
smedina@math.uc3m.es
120. Mendes, Ana. Instituto Politécnico de Leiria. Portugal.
aimendes@estg.ipleiria.pt
121. Milson, Robert. Dalhousie University. Canada.
rmilson@dal.ca
86
CHAPTER 9. LIST OF PARTICIPANTS
122. Minguez Ceniceros, Judit. Universidad de La Rioja. Spain.
judit.minguez@unirioja.es
123. Moale Johannes, Ionela. Kepler University Linz. Austria.
Ionela.Moale@risc.jku.at, Ionela.Moale@jku.at
124. Mohamed, Gaied. ISITCom, Hammam Sousse. Tunisia.
mohamed.gaied@ipeim.rnu.tn, gaied_mohamed@yahoo.fr
125. Montaner, Lavedán. Universidad de Zaragoza. Spain.
montaner@unizar.es
126. Moreno-Balcázar, Juan José. Universidad de Almerı́a. Spain.
balcazar@ual.es
127. Nagao, Taro. Nagoya University. Japan.
nagao@math.nagoya-u.ac.jp
128. Neuschel, Thorsten. University of Trier. Germany.
neuschel@uni-trier.de
129. Nevai, Paul. Ohio State University. USA.
nevai@math.ohio-state.edu
130. Ndayiragije, François. Katholieke Universiteit Leuven. Belgium.
ndayiragijefrancois@yahoo.fr
131. Njåstad, Olav. Norwegian University of Science and Technology. Norway.
Olav.Njastad@math.ntnu.no
132. Nojoua, Barhoumi. Preparaty instutit of ingineer Monastir. Tunisia.
133. Nowak, Rafal. University of Wroclaw. Poland.
rno@cs.uni.wroc.pl
134. Öffner, Philipp. TU Braunschweig. Germany.
p.oeffner@tu-bs.de
135. Oktay, Burcin. Balikesir University. Turkey.
burcinokt@gmail.com
136. Ong, Darren. Rice University. USA.
darren.ong@rice.edu
137. Orive, Ramón. Universidad de La Laguna. Spain.
rorive@ull.es
138. Oulad Yakhlef, Hossain. Universidad Carlos III de Madrid. Spain. houlad@math.uc3m.es
139. Paiva, Anabela. Universidade da Beira Interior. Portugal.
apaiva@mat.ubi.pt
87
140. Park, Choonkil. Hanyang University. South Korea.
baak@hanyang.ac.kr, cgpark@cnu.ac.kr
141. Pedersen, Henrik. University of Copenhagen - Faculty of Life Sciences. Denmark.
henrikp@dina.kvl.dk
142. Peña, Ana. Universidad de Zaragoza. Spain.
anap@unizar.es
143. Pérez, Carlos. Universidad de Vigo. Spain.
carlosp@edu.xunta.es, carlos@uvigo.es
144. Pérez, Dilcia. Universidad Centroccidental Lisandro Alvarado. Venezuela.
dperez@ucla.edu.ve, dilciaorama@gmail.com
145. Pérez Riera, Mario. Universidad de Zaragoza. Spain.
mperez@unizar.es
146. Pérez-Sinusı́a, Ester. University of Zaragoza. Spain.
ester.perez@unizar.es
147. Pérez, Teresa E. Universidad de Granada. Spain.
tperez@ugr.es
148. Pérez Valero, Marı́a Francisca. Universidad Carlos III de Madrid. Spain.
mpvalero@math.uc3m.es
149. Perdomo-Pı́o, Francisco. La Laguna University. Spain.
fjppio@ull.es
150. Pestana, Domingo. Universidad Carlos III de Madrid. Spain.
dompes@math.uc3m.es
151. Pestano-Gabino, Celina. La Laguna University. Spain.
cpestano@ull.es
152. Petronilho, José Carlos. Universidade de Coimbra. Portugal.
josep@mat.uc.pt
153. Pieter, Tibboel. Katholieke Universiteit Leuven. Belgium.
Pieter.Tibboel@wis.kuleuven.be
154. Pijeira Cabrera, Héctor. Universidad Carlos III de Madrid. Spain.
hpijeira@math.uc3m.es
155. Piñar, Miguel A. Universidad de Granada. Spain.
mpinar@ugr.es
156. Pirani, Vanessa. Universidade Estadual Paulista. Brazil.
botta@fct.unesp.br
157. Pizón Cortés, Natalia Camila. Universidad Carlos III de Madrid. Spain.
npinzon@math.uc3m.es, ncpinzonco@yahoo.es
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CHAPTER 9. LIST OF PARTICIPANTS
158. Portilla, Ana. Saint Louis University (Madrid campus). Spain.
ana.portillaferreira@gmail.com
159. Quintana, Yamilet. Universidad Simón Bolivar. Venezuela.
yquintana@usb.ve
160. Rafaeli, Fernando. Universidade Estadual Paulista. Brazil.
rafaeli@fct.unesp.br
161. Rakhmanov, Evguenii. University of South Florida. USA.
rakhmano@shell.cas.usf.edu
162. Ramos-López, Darı́o. University of Almerı́a. Spain.
dariorl@gmail.com
163. Rebocho, Maria das Neves. CMUC. Portugal.
mneves@mat.ubi.pt
164. Rehouma, Abdel Hamid. Central University of Eloued. Algeria.
mathsrah@hotmail.com
165. Rezola, Maria Luisa. Universidad de Zaragoza. Spain.
rezola@unizar.es
166. Rivero Castillo, Daniel. Universidad Carlos III de Madrid. Spain.
darivero@math.uc3m.es
167. Rocha Álvarez, Ignacio. Universidad Politécnica de Madrid. Spain.
igalvar@euitt.upm.es
168. Rodrigues, M. Manuela. University of Aveiro. Portugal.
mrodrigues@ua.pt
169. Rodrı́guez Garcı́a, José M. Universidad Carlos III de Madrid. Spain.
jomaro@math.uc3m.es
170. Román, Pablo. Katholieke Universiteit Leuven. Belgium.
roman@mate.uncor.edu, PabloManuel.Roman@wis.kuleuven.be
171. Romera, Elena. Universidad Carlos III de Madrid. Spain.
eromera@math.uc3m.es
172. Roncal, Luz. Universidad de La Rioja. Spain.
luz.roncal@unirioja.es
173. Saff, Edward. Vanderbilt University. USA.
Ed.Saff@Vanderbilt.Edu
174. Sánchez Dehesa, Jesús. Universidad de Granada. Spain.
dehesa@ugr.es
175. Sánchez Lara, Joaquı́n. Universidad de Granada, Spain.
jslara@ugr.es
89
176. Sánchez Moreno, Pablo. Universidad de Granada. Spain.
pablos@ugr.es
177. Sánchez Ruiz, Jorge. Universidad Carlos III de Madrid. Spain.
jsanchez@math.uc3m.es
178. Sastre, M. Asunción. Universidad Politécnica de Madrid. Spain.
masastre@fi.upm.es
179. Segura, Javier. Universidad de Cantabria. Spain.
javier.segura@unican.es
180. Schnieder, Jörn. University of Luebeck. Germany.
varif@web.de, schniede@math.uni-luebeck.de
181. Sevinik Adiguzel, Rezan. Middle East Technical University. Turkey.
rezan@metu.edu.tr
182. Sfaxi, Ridha. Higher Institute of Management Gabes Tunisia. Tunisia.
ridhasfaxi@yahoo.fr
183. Shayanfar, Nikta. K.N. Toosi University of Technology. Iran.
nikta.shayanfar@gmail.com
184. Simanek, Brian. California Institute of Technology. USA.
bsimanek@caltech.edu
185. Simon, Barry. California Institute of Technology. USA.
bsimon@caltech.edu
186. Smet, Christophe. Katholieke Universiteit Leuven. Belgium.
christophe@wis.kuleuven.be
187. Soria-Lorente, Anier. Universidad Carlos III de Madrid. Spain.
aslorent@math.uc3m.es
188. Sri Ranga, Alagacone. Universidade Estadual Paulista. Brazil.
ranga@ibilce.unesp.br
189. Stahl, Herbert. Technische Fachhochschule Berlin. Germany.
stahl@tfh-berlin.de
190. Stylianopoulos, Nikos. University of Cyprus. Cyprus.
nikos@ucy.ac.cy
191. Suárez Rodriguez, Carmen. Universidad de Vigo. Spain.
csuarez@uvigo.es
192. Temme, Nico. Centrum Wiskunde & Informatica. The Netherlands.
Nico.Temme@cwi.nl
193. Torrano, Emilio. Universidad Politécnica de Madrid. Spain.
emilio@fi.upm.es
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CHAPTER 9. LIST OF PARTICIPANTS
194. Totik, Vilmos. University of Szeged and University of South Florida. Hungary-USA.
totik@math.usf.edu, totik@math.u-szeged.hu
195. Tourı́s Lojo, Eva. Universidad Autónoma de Madrid. Spain.
eva.touris@uam.es
196. Tyaglov, Mikhail. Technische Universität Berlin. Germany.
tyaglov@gmail.com
197. Van Assche, Walter. Katholieke Universiteit Leuven. Belgium.
walter@wis.kuleuven.be
198. van Doorn, Erik. University of Twente. The Netherlands.
e.a.vandoorn@utwente.nl, e.a.vandoorn@kpnmail.nl
199. van de Bult, Fokko. California Institute of Technology. USA.
vdbult@caltech.edu, fjvdbult@gmail.com
200. van Pruijssen, Maarten. Radboud Universiteit, FNWI, IMAPP, Wiskunde. The Netherlands.
m.vanpruijssen@math.ru.nl
201. Varona, Juan Luis. Universidad de La Rioja. Spain.
jvarona@unirioja.es
202. Velázquez, Luis. Universidad de Zaragoza. Spain.
velazque@unizar.es
203. Verde-Star, Luis. Universidad Autonoma Metropolitana. Mexico.
verde@xanum.uam.mx, verde@star.izt.uam.mx
204. Viazovska, Maryna. Max Planck Institute for Mathematics. Germany.
viazovsk@mpim-bonn.mpg.de, viazovska@gmail.com
205. Vidunas, Raimundas. Kobe University. Japan.
vidunas@math.kobe-u.ac.jp
206. Vieira, Nelson. University of Porto. Portugal.
nvieira@fc.up.pt
207. Vinet, Luc. Université de Montreal. Canada.
luc.vinet@umontreal.ca
208. Vollrath, Antje. TU Braunschweig, AG Partielle Differentialgleichungen. Deutschland.
a.vollrath@tu-bs.de
209. Wielonsky, Franck. Université de Provence. France.
wielonsk@cmi.univ-mrs.fr
210. Wong, Roderick. City University of Hong Kong. China.
rscwong@cityu.edu.hk
211. Yattselev, Maxim. University of Oregon. USA.
maximy@uoregon.edu
91
212. Yuditskii, Peter. University of Linz. Austria.
Petro.Yudytskiy@jku.at
213. Zarzo Altarejos, Alejandro. Universidad Politécnica de Madrid. Spain.
azarzo@etsii.upm.es
214. Zhang, Lun. Katholieke Universiteit Leuven. Belgium.
lun.zhang@wis.kuleuven.be, lawrenfdu@hotmail.com
215. Zudilin, Wadim. The University of Newcastle. Australia.
wadim.zudilin@newcastle.edu.au, wzudilin@gmail.com
10
Campus Map
93
Room A (5.1.A01), Auditorium – Room B (4.1.E01) – Room C (4.1.E03) – Room D (4.1.E04)