Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 294134, 19 pages
doi:10.1155/2011/294134
Research Article
Jacobi-Sobolev Orthogonal Polynomials:
Asymptotics for N-Coherence of Measures
Bujar Xh. Fejzullahu1 and Francisco Marcellán2
1
2
Faculty of Mathematics and Sciences, University of Prishtina, Mother Teresa 5, 10000 Prishtina, Kosovo
Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid,
Avenida de la Universidad, 30, 28911 Leganes, Spain
Correspondence should be addressed to Francisco Marcellán, pacomarc@ing.uc3m.es
Received 24 November 2010; Accepted 7 March 2011
Academic Editor: Alexander I. Domoshnitsky
Copyright q 2011 B. Xh. Fejzullahu and F. Marcellán. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Let us introduce the Sobolev-type inner product f, g f, g1 λf , g 2 , where λ > 0 and
1
1
f, g1 −1 f xgx1 − xα 1 xβ dx, f, g2 −1 f xgx1 − xα1 1 xβ1 / M
k1 |x −
Nk
i
i
M
f
ξ
g
ξ
,
with
α,
β
>
−1,
|ξ
|
>
1,
and
M
>
0,
for
all
k, i. A
ξk |Nk 1 dx M
k,i
k
k
k
k,i
k1
i0
Mehler-Heine-type formula and the inner strong asymptotics on −1, 1 as well as some estimates
for the polynomials orthogonal with respect to the above Sobolev inner product are obtained.
Necessary conditions for the norm convergence of Fourier expansions in terms of such Sobolev
orthogonal polynomials are given.
1. Introduction
For a nontrivial probability measure σ, supported on −1, 1, we define the linear space
Lp dσ of all measurable functions f on −1, 1 such that fLp dσ < ∞, where
fLp dσ ⎧
1/p
1 ⎪
p
⎪
⎪
⎪
,
fx dσx
⎨
if 1 ≤ p < ∞,
⎪
⎪
⎪
⎪
⎩ess supfx,
if p ∞.
−1
−1<x<1
1.1
Let us now introduce the Sobolev-type spaces see, e.g., 1, Chapter 3 in a more general
framework
2
Journal of Inequalities and Applications
W
N,p
p
p
f : fW N,p fLp
p
dμα,β λf Lp
Nk
M
p
i1
Mk,i f
ξk < ∞ ,
dνα,β 1 ≤ p < ∞,
1.2
k1 i0
W N,∞
f : fW N,∞ max fL∞ dμα,β , λf L∞ dνα,β < ∞ ,
where λ > 0 and dμα,β x 1 − xα 1 xβ dx, dνα,β x 1 − xα1 1 xβ1 / M
k1 |x −
Nk 1
dx with α, β > −1, |ξk | > 1, and Mk,i > 0, for all k, i. We denote by N the vector of
ξk |
dimension M with components N1 , . . . , NM .
Let f and g in W N,2 . We can introduce the Sobolev-type inner product
f, g f, g 1 λ f , g 2 ,
1.3
where λ > 0 and
f, g 2 1
f, g
1
1
−1
fxgx1 − xα 1 xβ dx,
Nk
M
1 − xα1 1 xβ1
fxgx M
dx
Mk,i f i ξk g i ξk ,
Nk 1
−1
k1 i0
k1 |x − ξk |
1.4
1.5
where λ > 0, α, β > −1, |ξk | > 1, and Mk,i > 0, for all k, i. In the sequel, we will assume that
ξk < −1, and, therefore, |x − ξk | x − ξk for all k 1, 2, . . . , M, and −1 ≤ x ≤ 1.
Using the standard Gram-Schmidt method for the canonical basis xn n≥0 in the linear
space of polynomials, we obtain a unique sequence up to a constant factor of polynomials
α,β,N
Qn
n≥0 orthogonal with respect to the above inner product. In the sequel, they will called
Jacobi-Sobolev orthogonal polynomials.
For M 1 and N1 0, the pair of measures dμα,β , dνα,β M1,0 δξ is a 0-coherent pair,
studied in 2–4 see also 5 in a more general framework. In 6, the authors established
the distribution of the zeros of the polynomials orthogonal with respect to the above Sobolev
inner product 1.3 when M 1 and N1 0. Some results concerning interlacing and
separation properties of their zeros with respect to the zeros of Jacobi polynomials are also
obtained assuming we are working in a coherent case. More recently, for a noncoherent pair
of measures, when α β, M 2, N1 N2 0, and ξ1 −ξ2 , the distribution of zeros of
the corresponding Sobolev orthogonal polynomials as well as some asymptotic results more
precisely, inner strong asympttics, outer relative asymptotics, and Mehler-Heine formulas
for these sequences of polynomials are deduced in 7–9. In the Jacobi case, some analog
problems have been considered in 10, 11.
The aim of this contribution is to study necessary conditions for W N,p -norm
convergence of the Fourier expansion in terms of Jacobi-Sobolev orthogonal polynomials.
In order to prove it, we need some estimates and strong asymptotics for the polynomials
α,β,N
α,β,N
x as well as for their derivatives Q n
x. A Mehler-Heine-type formula, inner
Qn
strong asymptotics, upper bounds in −1, 1, and W N,p norms of Jacobi-Sobolev orthonormal
Journal of Inequalities and Applications
3
polynomials are obtained. Thus, we extend the results of 10 for generalized N-coherent
pairs of measures.
The structure of the manuscript is as follows. In Section 2, we give some basic
properties of Jacobi polynomials that we will use in the sequel. In Section 3, an algebraic
α,β,N
and Jacobi orthonormal
relation between the sequences of polynomials Qn
M n≥0
polynomials is stated. It involves N 1 where N k1 Nk 1 consecutive terms of
such sequences in such a way that we obtain a generalization of the relations satisfied in
α,β,N
the coherent case. Upper bounds for the polynomials Qn
x and their derivatives in
−1, 1 are deduced. The inner strong asymptotics as well as a Mehler-Heine-type formula
are obtained. Finally, the asymptotic behavior of these polynomials with respect to the
W N,p norm is studied. In Section 4, necessary conditions for the convergence of the Fourier
expansions in terms of the sequence of Jacobi-Sobolev orthogonal polynomials are presented.
Throughout this paper, positive constants are denoted by c, c1 , . . . and they may vary
at every occurrence. The notation un ∼
vn means that the sequence un /vn converges to 1 and
notation un ∼ vn means c1 un ≤ vn ≤ c2 un for sufficiently large n.
2. Preliminaries
α,β
For α, β > −1, we denote by pn n≥0 the sequence of Jacobi polynomials which are
orthonormal on −1, 1 with respect to the inner product
f, g 1 1
−1
fgdμα,β .
2.1
We will denote by kπn the leading coefficient of any polynomial πn x, and πn x kπn −1 πn x. Now, we list some properties of the Jacobi orthonormal polynomials which
we will use in the sequel.
α,β
Proposition 2.1. (a) The leading coefficient of pn
k
α,β
pn
1
n
2
2n α β
n
×
is (see [12, formulas (4.3.4) and (4.21.6)])
−1/2
2αβ1 Γn α 1Γ n β 1
.
2n α β 1 Γn 1Γ n α β 1
2.2
b The derivatives of Jacobi polynomials satisfy (see [12, formula (4.21.7)])
α1,β1
d α,β
pn x n n α β 1 pn−1
x.
dx
2.3
c For α, β ≥ −1/2, and q max{α, β}
α,β α,β max pn x pn a ∼
cnq1/2 ,
−1≤x≤1
where a 1 if q α and a −1 if q β (see [12, Theorem 7.32.1]).
2.4
4
Journal of Inequalities and Applications
α,β
d For the polynomials pn
Theorem 1]):
, we get the following estimate (see [12, formula (7.32.6)], [13,
α,β pn x ≤ c1 − x−α/2−1/4 1 x−β/2−1/4 ,
2.5
where x ∈ −1, 1 and α, β ≥ −1/2.
e Mehler-Heine formula (see [12, Theorem 8.1.1])
α,β
lim n−α−1/2 pn
cos
n→∞
z
n
2α−β/2 z−α Jα z,
2.6
where α, β are real numbers and Jα z is the Bessel function of the first kind. This formula
holds locally uniformly, that is, on every compact subset of the complex plane.
α,β
f Inner strong asymptotics. For pn
[12, Theorem 8.21.8])
α,β
pn
x, when x ∈ −1 , 1 − and 0 < < 1, we get (see
α,β
x rn 1 − x−α/2−1/4 1 x−β/2−1/4 cos kθ γ O n−1 ,
2.7
α,β
1/2
where x cos θ, k n α β 1/2, γ −α 1/2π/2, and rn ∼
2/π .
g For α, β > −1, τ max{α, β}, and 1 ≤ p ≤ ∞ (see [12, p.391. Exercise 91], [14, (2.2)],
[15, Theorem 2]),
α,β
pn Lp dμα,β ⎧
⎪
c,
⎪
⎪
⎨
1/p
∼
log n
,
⎪
⎪
⎪
⎩ τ1/2−2τ2/p
,
n
if 2τ > pτ − 2 p/2,
if 2τ pτ − 2 p/2,
2.8
if 2τ < pτ − 2 p/2.
Let {sn x}∞
n0 be the sequence of orthonormal polynomials with respect to the inner
product 1.5, and let
ΠN x 1 N
bk Tk x
2.9
k1
√
Nk 1
be the N-th polynomial orthonormal with respect to dx/π 1 − x2 M
, where
k1 x − ξk M
N k1 Nk 1 and Tk x cos kθ, x cos θ, are the Tchebychev polynomials of the first
kind.
Proposition 2.2 16, Lemma 2.1. For n ≥ N, there exist constants An,i such that
sn x N
i0
α1,β1
An,i pn−i
x
2.10
Journal of Inequalities and Applications
5
and limn → ∞ An,i Ai , where
1
A0 ,
N
2 bN
bi
Ai ,
2 N bN
1 ≤ i ≤ N.
2.11
Next, we will consider the polynomials
vn1 x N
α,β
an,i pn−i1 x,
2.12
i0
where an,i An,i n 1n α β 2/n − i 1n − i α β 2, 0 ≤ i ≤ N. Notice that
N
ΠN 1
an,i ∼
.
2 N bN
i0
2.13
Taking√ into account that the zeros of the polynomial ΠN x orthogonal with respect to
Nk 1
on the interval −1, 1 are real, simple, and located in −1, 1,
dx/π 1 − x2 M
k1 x − ξk 0/
ΠN −1. Therefore, N
0 for n large enough.
we have ΠN 1 /
i0 an,i /
On the other hand, using b in Proposition 2.1, we have
vn1
x n 1 n α β 2 sn x.
2.14
From Proposition 2.1 and 2.12, we get the following.
Proposition 2.3. a For α, β ≥ −1/2, and q max{α, β},
max |vn x| |vn a| ∼
cnq1/2 ,
−1≤x≤1
2.15
where a 1 if q α and a −1 if q β.
b When x ∈ −1, 1 and α, β ≥ −1/2, we get the following estimate for the polynomials vn :
|vn x| ≤ c1 − x−α/2−1/4 1 x−β/2−1/4 .
2.16
c Mehler-Heine type formula. We get
z
ΠN 1 −α
2α−β/2 lim n−α−1/2 vn cos
z Jα z,
n→∞
n
2 N bN
2.17
where α, β are real numbers, and Jα z is the Bessel function of the first kind. This formula holds
locally uniformly, that is, on every compact subset of the complex plane.
6
Journal of Inequalities and Applications
d Inner strong asymptotics. When x ∈ −1 , 1 − and 0 < < 1, we get
vn x 1 − x−α/2−1/4 1 x−β/2−1/4
N
α,β
an−1,i rn−i cos k − iθ γ O n−1 ,
2.18
i0
α,β
1/2
where x cos θ, k n α β 1/2, γ −α 1/2π/2, and rn ∼
2/π .
e For α, β > −1, 1 ≤ p ≤ ∞, and 1 ≤ j ≤ n,
α,β
vj Lp dμα,β ≤ cpj
α,β
Lp dμα,β ≤ cpn
Lp dμα,β .
2.19
3. Asymptotics of Jacobi-Sobolev Orthogonal Polynomials
α,β,N
Let {Qn
x}∞
n0 denote the sequence of polynomials orthogonal with respect to 1.3
normalized by the condition that they have the same leading coefficient as vn x, that is,
α,β,N
α,β
an−1,0 kpn .
kQn
α,β,N
and vn x holds.
The following relation between Qn
Proposition 3.1. For α, β > −1,
α,β,N
vn x Qn
x N
α,β,N
n
αn−k Qn−k x,
n ≥ 1,
3.1
k1
where, for 1 ≤ k ≤ N,
α,β α,β,N a
pn−i , Qn−k
n−1,i
ik
N
n
αn−k n
1
α,β,N 2
Qn−k W N,2
.
3.2
n
Moreover, |αn−N | O1/n2 and |αn−k | O1/n for 1 ≤ k < N.
α,β,N n
}k0
Proof. Expanding vn x with respect to the basis {Qk
polynomials with degree at most n, we get
α,β,N
vn x Qn
x n
α,β,N
n
αn−k Qn−k
x,
of the linear space of
3.3
k1
where, for k 1, . . . , n,
n
αn−k
α,β,N
vn , Qn−k
α,β,N
Qn−k
α,β,N
, Qn−k
.
3.4
Journal of Inequalities and Applications
7
For k 1, . . . , n,
α,β,N
vn , Qn−k
α,β,N
vn , Qn−k
1
1
α,β,N
−1
N
α,β,N λ n n α β 1 sn−1 , Qn−k
vn xQn−k
1
an−1,i
i0
xdμα,β x
α,β
−1
α,β,N
pn−i xQn−k
2
3.5
xdμα,β x.
Therefore,
α,β,N
vn , Qn−k
0,
k > N,
1
N
α,β,N
α,β
α,β,N
vn , Qn−k
an−1,i
pn−i xQn−k xdμα,β x,
−1
ik
3.6
1 ≤ k ≤ N.
As a conclusion,
n
αn−k 0,
3.7
k > N,
α,β α,β,N pn−i , Qn−k
a
n−1,i
ik
N
n
αn−k α,β,N 2
Qn−k W N,2
1
,
1 ≤ k ≤ N.
3.8
Using the extremal property for monic orthogonal polynomials with respect to the corresponding norm see 12, Theorem 3.1.2,
α,β 2
L2 dμα,β pn
!
inf p2L2
dμα,β "
: deg p n, p monic ,
3.9
we get
α,β,N 2
W N,2
n
Q
λ
≥ M
k1 1
≥ M
Nk 1
− ξk λn2
k1 1
α,β,N 2
L2 dμα1,β1 # n
Q
α1,β1 2
pn−1
L2 dμα1,β1 .
Nk 1
− ξk 3.10
8
Journal of Inequalities and Applications
Thus,
α,β,N 2
W N,2
Qn
α,β,N 2 α1,β1 −2
k pn−1
≥ cn2 k Qn
≥ cn2 .
3.11
Finally, from 3.8, we find that
1
n |an−1,N an−N−1,0 |
O 2 ,
αn−N α,β,N 2
n
Qn−N W N,2
3.12
and from Schwarz inequality,
$ α,β,N α,β,N α,β,N
α,β α,β,N ≤ Qn−k W N,2 .
Qn−k , Qn−k
pn−i , Qn−k
≤
1
3.13
1
Thus,
1
n ,
αn−k O
n
1 ≤ k < N.
3.14
α,β,N
Using 3.1 in a recursive way, we get the representation of the polynomial Qn
terms of the elements of the sequence {vn x}∞
n0 . More precisely we get the following.
in
Proposition 3.2. For α, β > −1, it holds that
α,β,N
Qn
x n
m
3.15
b1,n vn−m x,
m0
0
1
n
m
m−1 n−m1
αn−k−m1
where b1,n 1, bk,n −αn−k , and bk,n −b1,n
m
m−1
bk1,n , k 1, 2, . . . , N, m 2, 3, . . . , n.
m
Moreover, |bk,n | O1/n for m 1, 2, . . . , N − 1, and |bk,n | O1/n2 for m N, N 1, . . . , n.
0
1
n
m
m−1 n−m1
αn−k−m1
Proof. Let denote by b1,n 1, bk,n −αn−k , and bk,n −b1,n
m 2, 3, . . . , n. First, we prove that
α,β,N
Qn
x l
m
b1,n vn−m x m0
N
l1 α,β,N
bk,n Qn−k−l x,
k1
α,β,N
where l 0, 1, . . ., and, by convention, Q−s
x 0, s 1, 2, . . ..
m−1
bk1,n , k 1, 2, . . . , N,
3.16
Journal of Inequalities and Applications
9
We will prove 3.16 by induction. When l 0, it is a trivial result. On the other hand,
applying 3.1 in a recursive way, we get
α,β,N
Qn
N
x vn x −
n
α,β,N
αn−k Qn−k
x
k1
&
N
N
α,β,N
α,β,N
n−1
n
vn−1 x − αn−k−1 Qn−k−1 x − αn−k Qn−k x
%
n
αn−1
vn x −
k1
1
N
1
m
b1,n vn−m x − b1,n
m0
1
k2
α,β,N
n−1
αn−k−1 Qn−k−1 x
k1
m
b1,n vn−m x m0
N
3.17
N−1
α,β,N
1
bk1,n Qn−k−1 x
k1
α,β,N
2
1
α,β,N
bk,n Qn−k−1 x − bN1,n Qn−N−1 x.
k1
1
Taking into account 3.7, we have bN1,n 0. Thus, 3.16 follows for l 1. Now, we assume
3.16 holds for l ≥ 1. Again, from 3.1,
N
l1
α,β,N
α,β,N
l1
bk,n Qn−k−l x b1,n Qn−l−1 x k1
N
l1
α,β,N
bk,n Qn−k−l x
k2
%
l1
b1,n
&
N
N−1
l1 α,β,N
α,β,N
n−l−1
vn−l−1 x − αn−k−l−1 Qn−k−l−1 x bk1,n Qn−k−l−1 x
k1
l1
b1,n vn−l−1 x N
l2
k1
α,β,N
l1
α,β,N
bk,n Qn−k−l−1 x − bN1,n Qn−N−l−1 x.
k1
l1
Now, we prove that bk,n
l
n−l
l
3.18
l1
0 for k > N. For l 0, this follows from 3.7. Since bk,n
n−l
−b1,n αn−k−l bk1,n and αn−k−l 0, for k > N the statement follows by induction. Thus, 3.16
holds for l 1. Now taking l n in 3.16, we get 3.15.
l
l
Finally, we prove that |bk,n | O1/n for l 1, 2, . . . , N − 1, and |bk,n | O1/n2 for
l N, N 1, . . . , n. First, the following inequality holds:
l bk,n ⎧ 1
⎪
⎪
,
O
⎪
⎪
n
⎪
⎪
⎪
⎪
⎨ 1
⎪O nn − l 1 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0,
if 1 ≤ k ≤ N − l,
if N − l 1 ≤ k ≤ N,
if k > N,
3.19
10
Journal of Inequalities and Applications
l 1, 2, . . . , N. Indeed, for l 1, 3.19 follows from Proposition 3.1 and 3.7. Now, we
assume that the relation 3.19 holds for l ≥ 1. Thus, for 1 ≤ k ≤ N − l − 1,
1
l ,
b1,n O
n
1
l ,
bk1,n O
n
1
n−l ,
αn−l−k O
n−l
3.20
for N − l ≤ k ≤ N − 1
1
l ,
b1,n O
n
l bk1,n O
1
,
nn − l 1
1
n−l ,
αn−l−k O
n−l
3.21
for k N
1
l ,
O
b1,n n
l bk1,n 0,
n−l αn−l−k O
1
n − l2
,
3.22
and for k > N
l bk1,n 0,
n−l αn−l−k 0.
3.23
Therefore, from
l1
l
n−l
l
bk,n −b1,n αn−k−l bk1,n ,
3.24
l
the relation 3.19 holds for l 1. As consequence, |bk,n | O1/n for l 1, 2, . . . , N − 1 and
1 ≤ k ≤ N.
l
Now, we will prove by induction that |bk,n | O1/n2 for l ≥ N and 1 ≤ k ≤ N.
l
The case l N follows from 3.19. We assume that |bk,n | O1/n2 for l ≥ N and
1 ≤ k ≤ N. For 1 ≤ k ≤ N − 1,
1
l b1,n O 2 ,
n
1
l bk1,n O 2 ,
n
1
n−l ,
αn−l−k O
n−l
3.25
and for k N
1
l b1,n O 2 ,
n
l bk1,n 0,
1
n−l .
αn−l−k O
n−l
3.26
Therefore, from
l1
l
n−l
l
bk,n −b1,n αn−k−l bk1,n ,
3.27
the statement holds for l 1.
Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.
Journal of Inequalities and Applications
11
α,β,N
Proposition 3.3. a For the polynomials Qn
, we get
α,β,N x ≤ c1 − x−α/2−1/4 1 x−β/2−1/4 ,
Qn
3.28
where x ∈ −1, 1, and α, β ≥ −1/2.
α,β,N
(b) For the polynomials Q n
, we get
α,β,N x ≤ cn1 − x−α/2−3/4 1 x−β/2−3/4 ,
Q n
3.29
where x ∈ −1, 1, and α, β > −1.
Proof. a Using Proposition 3.2, we have
N−1
n
1 1
α,β,N x ≤ |vn x| O
|vn−m x| O 2
|vn−m x|.
Qn
n m1
n mN
3.30
Therefore, from Proposition 2.3b, the statement follows immediately.
On the other hand, taking into account Proposition 2.1d, Proposition 2.2, 2.14, and
3.15, the proof of b can be done in a similar way.
α,β,N
Now, we show that, like for the classical Jacobi polynomials, the polynomial Qn
attains its maximum in −1, 1 at the end-points. More precisely,
x
Proposition 3.4. a For α, β ≥ −1/2, and q max{α, β}
α,β,N α,β,N max Qn
x Qn
a ∼ nq1/2 ,
−1≤x≤1
3.31
where a 1 if q α and a −1 if q β.
(b) For α, β > −1 and q max{α, β}
α,β,N α,β,N max Q n
x Q n
b ∼ nq5/2 ,
−1≤x≤1
3.32
where b 1 if q α and b −1 if q β.
Proof. Here, we will prove only the case when α ≥ β. The case when β ≥ α can be done in a
similar way.
a From Proposition 2.3a,
|vn−m x| ≤ cnα1/2 ,
m 0, 1, . . . , n,
3.33
for x ∈ −1, 1 and α ≥ β ≥ −1/2. Therefore, according to 3.30,
α,β,NN x ≤ cnα1/2 ,
Qn
3.34
12
Journal of Inequalities and Applications
for x ∈ −1, 1 and α ≥ β ≥ −1/2. From Proposition 3.1, we get
α,β,N x |vn x| − O nα−1/2 .
Qn
3.35
Finally, from Proposition 2.3a, the statement follows.
b Taking into account Proposition 2.1c, Proposition 2.2, 2.14, 3.1, and 3.15, we
can conclude the proof in the same way as we did in a.
Corollary 3.5. For α, β ≥ −1/2,
α,β,N
cos θ ≤ cA n, α, β, θ ,
Qn
3.36
α,β,N
cos θ ≤ cnA n, α 1, β 1, θ ,
Q n
3.37
and for α, β > −1,
where
⎧
−β−1/2
−α−1/2
⎪
,
−
θ
θ
π
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ α1/2
,
A n, α, β, θ n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩nβ1/2 ,
if
π −c
c
≤θ≤
,
n
n
if 0 ≤ θ ≤
if
c
,
n
3.38
π −c
≤ θ ≤ π.
n
Proof. The inequality
nα1/2 ≤ cθ−α−1/2
3.39
nβ1/2 ≤ cπ − θ−β−1/2 ,
3.40
holds for θ ∈ 0, c/n, as well as
for θ ∈ π − c/n, π. Therefore, from Propositions 3.3 and 3.4, the statement follows
immediately.
α,β,N
Next, we deduce a Mehler-Heine-type formula for Qn
Theorem 4.1 in 10.
α,β,N
and Q n
see
Proposition 3.6. Uniformly on compact subsets of C,
a
α,β,N
lim n−α−1/2 Qn
n→∞
ΠN 1 −α
z
2α−β/2 cos
z Jα z,
n
2 N bN
3.41
Journal of Inequalities and Applications
13
b
lim n
n→∞
−α−5/2
α,β,N
Q n
ΠN 1 −α−1
z
2α−β/2 z
Jα1 z,
cos
n
2 N bN
3.42
where α, β are real numbers, and Jα z is the Bessel function of the first kind.
Proof. To prove the proposition, we use the same technique as in 17.
a Multiplying in 3.1 by n 1−α−1/2 , we obtain
Vn z Yn z N
n
An−k Yn−k z,
n ≥ 1,
3.43
k1
α,β,N
where Yn z n 1−α−1/2 Qn
n
n
cos z/n, Vn z n 1−α−1/2 vn cos z/n and An−k n
n
αn−k n − k/n 1α1/2 , k 1, . . . , N. Moreover, |An−N | O1/n2 and |An−k | O1/n for
1 ≤ k < N.
Using the above relation in a recursive way as well as the same argument of
Proposition 3.2, we have
Yn z n
m
3.44
B1,n Vn−m z,
m0
0
m
m
where B1,n 1, |B1,n | O1/n for m 1, 2, . . . , N − 1, and |B1,n | O1/n2 for m N, N 1, . . . , n. Thus,
|Yn z| ≤
n m B1,n |Vn−m z|.
3.45
m0
On the other hand, from Proposition 2.3c, Vn n≥0 is uniformly bounded on compact subsets
of C. Thus, for a fixed compact set K ⊂ C, there exists a constant C, depending only on K,
such that when z ∈ K,
|Vn z| < C,
n ≥ 0.
3.46
Thus, the sequence Yn n≥0 is uniformly bounded on K ⊂ C. As a conclusion,
Yn z Vn z O n−1 ,
z ∈ K,
3.47
and from Proposition 2.3c, we obtain the result.
b Since we have uniform convergence in 3.41, taking derivatives and using a well
known property of Bessel functions of the first kind see 12, formula 1.71.5, we obtain
3.42.
α,β,N
Now, we give the inner strong asymptotics of Qn
on −1, 1.
14
Journal of Inequalities and Applications
Proposition 3.7. For x ∈ −1 , 1 − and 0 < < 1,
α,β,N
Qn
x 1 − x−α/2−1/4 1 x−β/2−1/4 ×
N
α,β
an−1,i rn−i cos k − iθ γ O n−1 ,
i0
3.48
α,β,N
Q n
cos θ n n α β 1 1 − x−α/2−3/4 1 x−β/2−3/4
×
N
α1,β1
An−1,i rn−i−1 cos k − iθ γ1 O1,
3.49
i0
where x cos θ, k n α β 1/2, γ −α 1/2π/2, γ1 −α 3/2π/2, and
α,β
1/2
rn ∼
2/π .
α,β,N
Proof. From Proposition 3.3a, the sequence Qn
subsets of −1, 1; thus, from Proposition 3.1,
α,β,N
Qn
x
n≥0 is uniformly bounded on compact
1
vn x O
.
n
3.50
Now, using Proposition 2.3d, the relation 3.48 follows.
Concerning 3.49, it can be obtained in a similar way by using Propositions 2.1f and
2.2, 2.14, Propositions 3.1 and 3.3b.
Now, we can give the sharp estimate for the Sobolev norms of the Jacobi-Sobolev
polynomials.
Proposition 3.8. For α ≥ β ≥ −1/2 and 1 ≤ p ≤ ∞,
α,β,N
Qn
W N,p
⎧
⎪
⎪
n,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1/p
,
∼ n log n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩nα5/2−2α4/p ,
if
4α 2
> p,
2α 3
if
4α 2
p,
2α 3
if
4α 2
< p.
2α 3
3.51
Proof. Clearly, if p ∞, then we get Proposition 3.4b. Thus, in the proof, we will assume
1 ≤ p < ∞. Since by Proposition 3.2 and2.14
N−1
n−1 i
di
di1
di
d
α,β,N
i
ξk ≤ cn i sn−1 ξk c i sn−m−1 ξk O n−1
i1 Qn
i sn−m−1 ξk ,
dx
dx
dx
dx
m1
mN
3.52
Journal of Inequalities and Applications
15
where k 1, . . . , M, i 0, 1, . . . , Nk , and di /dxi sn ξk are bounded because of the
orthonormality condition, we obtain
di1
α,β,N
ξk ≤ cn,
i1 Qn
dx
3.53
where k 1, . . . , M, and i 0, 1, . . . , Nk .
On the other hand, using 3.30, Minkowski’s inequality, and Proposition 2.3e, we
deduce
α,β,N
Qn
α,β
Lp dμα,β ≤ cpn
α1,β1
Lp dμα,β ≤ cnpn
Lp dμα1,β1 .
3.54
In the same way as above, we get
α,β,N
Q n
α1,β1
Lp dμα1,β1 ≤ cnpn
Lp dμα1,β1 .
3.55
Thus, from 3.53, 3.54, and 3.55, we have
α,β,N p
W N,p
Qn
≤
α,β,N p
Qn
Lp dμα,β α1,β1
≤ cnp pn
M
λ
k1 −1
α,β,N p
Q n
Lp dμα1,β1 Nk 1
− ξk p
di1
α,β,N
Mk,i i1 Qn
ξk dx
k1 i0
Nk
M
Lp dμα1,β1 .
3.56
Notice that the upper estimate in 3.54 and 3.55 can also be proved using the bounds for
Jacobi-Sobolev polynomials given in Corollary 3.5.
In order to prove the lower bound in 3.51 we will need the following.
Proposition 3.9. For α > −1 and 1 ≤ p < ∞,
α,β,N
Q n
Lp dμα1,β1 ⎧
⎪
⎪
⎪cn,
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1/p
≥ c cn log n
,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩nα5/2−2α4/p ,
if
4α 2
> p,
2α 3
if
4α 2
p,
2α 3
if
4α 2
< p.
2α 3
3.57
16
Journal of Inequalities and Applications
Proof. We will use a technique similar to 12, Theorem 7.34. According to 3.42,
π/2
0
p
α,β,N
θ2α3 Q n
cos θ dθ
ω/n
>
0
p
α,β,N
θ2α3 Q n
cos θ dθ
cn−2α−4
×
ω
ω
0
α,β,N
t p ∼ pα5/2−2α−4
cos
t2α3 Q n
dt cn
n 3.58
p
t2α3 t−α−1 Jα1 t dt cnpα5/2−2α−4
0
×
ω
t2α3−pα−p |Jα1 t|p dt.
0
On the other hand, from see 18, Lemma 2.1, if γ > −1 − pα and 1 ≤ p < ∞, we have
ω
tγ |Jα t|p dt ∼
0
⎧
⎪
⎪
⎨c,
⎪
⎪
⎩c log ω,
if γ <
p
,
2−1
p
if γ .
2−1
3.59
Thus, for 4α 2/2α 3 ≤ p and ω large enough, 3.57 follows.
Finally, from 3.49, we obtain
π/2
θ
0
π/2
p
p
α,β,N
α,β,N
θ2α3 Q n
cos θ dθ >
cos θ dθ ≥ cnp .
Q n
2α3 3.60
π/4
The proof of Proposition 3.9 is complete.
From 3.57, for α > −1 and 1 ≤ p < ∞,
α,β,N
Qn
W N,p
⎧
⎪
⎪
cn,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1/p
,
≥ c n log n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩nα5/2−2α4/p ,
Thus, using 3.56 and 3.61, the statement follows.
if
4α 2
> p,
2α 3
if
4α 2
p,
2α 3
if
4α 2
< p.
2α 3
3.61
Journal of Inequalities and Applications
17
4. Necessary Conditions for the Norm Convergence
The analysis of the norm convergence of partial sums of the Fourier expansions in terms
of Jacobi polynomials has been done by many authors. See, for instance, 19–21, and the
references therein.
α,β,N
Let qn
be the Jacobi-Sobolev orthonormal polynomials, that is,
α,β,N
qn
−1 α,β,N
α,β,N
W N,2
Qn
x Qn
x.
4.1
For f ∈ W N,1 , its Fourier expansion in terms of Jacobi-Sobolev orthonormal
polynomials is
∞
α,β,N
fkq
x,
k
4.2
k0
where
α,β,N fk
f, qk
,
k 0, 1, . . . .
4.3
Let Sn f be the n-th partial sum of the expansion 4.2
n
α,β,N
fkqk
Sn f, x x.
4.4
k0
Theorem 4.1. Let α ≥ β ≥ −1/2, and 1 ≤ p ≤ ∞. If there exists a constant c > 0 such that
Sn fW N,p ≤ cfW N,p ,
4.5
for every f ∈ W N,p , then p ∈ p0 , q0 with
q0 4α 2
,
2α 3
p0 4α 2
.
2α 5
4.6
Proof. For the proof, we apply the same argument as in 20. Assume that 4.5 holds. Then,
'
α,β,N '
α,β,N
'
'
qn
x'
' f, qn
W N,p
Sn f − Sn−1 fW N,p ≤ cfW N,p .
4.7
Consider the linear functionals
α,β,N α,β,N
qn
W N,p
Tn f f, qn
4.8
18
Journal of Inequalities and Applications
on W N,p . Hence, for every f in W N,p supn |Tn f| < ∞ holds. From the Banach-Steinhaus
theorem, this yields supn Tn < ∞. On the other hand, by duality see, for instance, 1,
Theorem 3.8, we have
α,β,N
Tn qn
α,β,N
W N,p qn
W N,q ,
4.9
xW N,q < ∞.
4.10
where p is the conjugate of q. Therefore,
α,β,N
supn qn
α,β,N
xW N,p qn
On the other hand, from 3.51, we obtain the Sobolev norms of Jacobi-Sobolev
orthonormal polynomials
α,β,N
qn
W N,p
⎧
⎪
c,
⎪
⎪
⎨
1/p
∼
log n
,
⎪
⎪
⎪
⎩ α3/2−2α4/p
,
n
if p < q0 ,
if p q0 ,
4.11
if p > q0 ,
for α ≥ β ≥ −1/2 and 1 ≤ p ≤ ∞. Now, from 4.11, it follows that the inequality 4.10 holds
if and only if p ∈ p0 , q0 .
The proof of Theorem 4.1 is complete.
Acknowledgments
The authors thank the referees for the careful revision of the manuscript. Their comments
and suggestions have contributed to improve substantially its presentation. The work of F.
Marcellán has been supported by Dirección General de Investigación, Ministerio de Ciencia
e Innovación of Spain, Grant no. MTM2009-12740-C03-01.
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