Proceedings of DWCAA12, Volume 6 · 2013 · Pages 62–73
Error Estimates for Polyharmonic Cubature Formulas
Hermann Render a · Ognyan Kounchev b
Abstract
In the present article we shall present basic features of a polyharmonic cubature formula of degree
s and corresponding error estimates. Main results are Markov-type error estimates for differentiable
functions and error estimates for functions f which possess an analytic extension to a sufficiently
large ball in the complex space Cd .
2000 AMS subject classification: 65D30, 32A35
Keywords: Numerical integration, cubature formula, Fourier-Laplace series.
1
Introduction
€ Š
Let C Rd be the set of all continuous complex-valued functions on the euclidean space Rd . A cubature formula C is a
€ Š
linear functional on C Rd of the form
C f := α1 f x 1 + .... + αN f x N .
(1)
The points x 1 , ..., x N are called nodes or knots and the coefficients α1 , ..., αN ∈ R the weights. A basic problem in numerical
analysis is to approximate integrals of the form
Z
f (x) dµ (x)
for a (signed) measure µ in the euclidean space Rd by suitable cubature formulas.
€ Š
An important characteristic of a cubature formula is exactness: the functional C is exact on a subspace U of C Rd
with respect to a measure µ if
Z
C f = f (x) dµ (x)
(2)
holds for all f ∈ U. If Us is the set of all polynomials Ps of degree ≤ s, and the cubature is exact on Us but not on Us+1 , we
say that C has order s. Exactness on the space Ps can be expressed by the identities
Z
C (x α ) =
x α dµ (x)
α
α
for each multi-index α = α1 , ..., αd ∈ N0d with |α| := α1 + ... + αd ≤ s where x α = x 1 1 ...x d d . In the theory of cubature
formula it is assumed that the moments
Z
x α dµ (x)
a
School of Mathematical Sciences, University College Dublin (Ireland)
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Str., 1113 Sofia (Bulgaria) and Interdisciplinary Center for
Complex Systems (IZKS), University of Bonn (Germany)
b
Render · Kounchev
63
for |α| ≤ s exist and that they can be explicitly calculated. The problem is to find constructive methods for determining
nodes and weights from this information. In particular, a cubature formula leads to a solution of the so-called truncated
moment problem. For a discussion of cubature formulas we refer to [26], [27], [29] and the recent survey [7].
In [15] and [18] we have introduced a new type of functional which approximates the integral
Z
f (x) dµ (x)
(3)
for a class of measures µ with support in the ball
¦
©
BR = x ∈ Rd : |x| < R
(4)
and continuous functions f : BR → C where R is a positive number or ∞, and
p
r = |x| = x 12 + .... + x d2
is the euclidean norm of x = x 1 , ..., x d ∈ Rd . The unit sphere will be denoted by
¦
©
Sd−1 := x ∈ Rd : |x| = 1 ,
and endowed with the rotation invariant measure dθ .
Our approach is based on the Fourier-Laplace series of the function f (x) . In order to make concepts simpler we shall
restrict our discussion in the introduction to the two-dimensional case where the Fourier-Laplace series is just the Fourier
series of a function. Hence we define the basis functions
1
Y0,0 (x) = Y0,0 (r cos t, r sin t) = p
(5)
2π
and
1
Yk,1 (x) = Yk,1 (r cos t, r sin t) = p r k cos kt
π
1 k
Yk,2 (x) = Yk,2 (r cos t, r sin t) = p r sin kt
π
(6)
(7)
for k ∈ N where N denotes the set of all natural numbers, and N0 := N ∪{0} . A point x ∈ R2 is written as x = (r cos t, r sin t)
where r is the radius of x and (cos t, sin t) is in the unit sphere. The Fourier coefficients of a continuous function f are
defined by
Z 2π
f k,` (r) =
f (r cos t, r sin t) · Yk,` (cos t, sin t) d t.
0
The Fourier series of the continuous function f : BR → C is defined by the formal expansion
ak
∞ X
X
f k,` (r) Yk,` (θ )
(8)
k=0 `=1
where a0 = 1 and ak = 2 for k ∈ N, and θ = (cos t, sin t) . It is easy to see that f k,` is a continuous function if f is continuous.
Furthermore, if f is infinitely differentiable in BR then the function
f k,` (r) r −k
is even (and infinitely differentiable), see [6]. Finally, if f is a polynomial then f k,` (r) r −k is a univariate polynomial in r 2 ,
see Section 2 for more details.
If f is sufficiently smooth then the Fourier series (8) converges absolutely and uniformly on compact subsets of BR to
the function f (x) and one obtains that
Z
ak Z
∞ X
X
f (x) dµ =
f k,` (r) Yk,` (θ ) dµ (x)
R2
k=0 `=1
=
ak
∞ X
X
k=0 `=1
Dolomites Research Notes on Approximation
R2
Z
f k,` (r) r −k Yk,` (x) dµ (x) .
R2
ISSN 2035-6803
Render · Kounchev
64
We shall now call a signed measure µ with support in BR ⊂ R2 pseudo-positive if the inequality
Z
h (|x|) Yk,` (x) dµ (x) ≥ 0
R2
holds for every non-negative continuous function h : [0, R] → [0, ∞) and for all k ∈ N0 , and ` = 1, ..., ak . By the Riesz
representation theorem there exist unique non-negative measures µk,` defined on [0, R] , which we call component measures,
such that
Z∞
Z
h (t) dµk,` (t) =
h (|x|) Yk,` (x) dµ
0
holds for all h ∈ C [0, R]. Using this notation we obtain
Z
ak Z
∞ X
X
f (x) dµ =
R2
k=0 `=1
R2
∞
f k,` (r) r −k dµk,` (r) .
0
In passing, we mention that radially symmetric measures are pseudo-positive.
The main idea in our approach is to use quadrature formulas to approximate the univariate integrals
Z∞
f k,` (r) r −k dµk,` (r) .
(9)
0
Thus we assume in our approach that the Fourier coefficients f k,` (r) are known. One may use Fast-Fourier Transform to
find approximations of f k,` and to combine these with our approach in order to find cubature formulas only involving the
function values of f – a topic which we want to consider in a future paper.
Next we want to discuss which kind of quadrature formulas for approximating (9) are useful. Due to the fact that
f k,` (r) r −k is an even function for smooth f we shall require that the quadrature formula is exact p
for all polynomials
of the form r 2 j for j = 0, ..., 2s − 1 where s a given natural number. By taking the transformation r this means that
the transformed quadrature formula should be exact for all polynomial t j for j = 0, ..., 2s − 1 – and here the classical
Gauß-Jacobi quadrature enters the game.
Our polyharmonic cubature formula is now defined in the following way: given a pseudo-positive measure µ we
ψ
consider the component measures µk,` (r) . Let µk,` be the image measure of µk,` for the transformation ψ : [0, ∞) → [0, ∞)
2
defined by ψ (r) = r , so
Z∞
Z∞
€p Š
ψ
−k
f k,` (r) r dµk,` (r) =
f k,`
t t −k/2 dµk,` (t) .
0
0
ψ
(s)
For the non-negative univariate measures µk,` we shall use the univariate Gauß-Jacobi quadratures νk,` of order 2s − 1 as
ψ
an approximation of µk,` . The polyharmonic cubature T (s) f of degree s is then defined by
ak Z ∞
∞ X
X
€p Š 1
(s)
T (s) f :=
f k,`
t t − 2 k dνk,` (t) .
k=0 `=1
0
The cubature formula T (s) will be defined at first only for polynomials: then the sum in the definition of T (s) f
a finite sum and no convergence questions occur. The cubature formula T (s) has the property that
Z
€ 2j
Š
(s)
T
|x| Yk,` (x) = |x|2 j Yk,` (x) dµ (x)
is actually
for all j = 0, ..., 2s − 1 and for all k ∈ N0 , ` = 1, ..., ak . This is equivalent to the functional T (s) being exact on the space of
all polynomials of polyharmonic order ≤ 2s.
In [15] we investigated the truncated moment problem for pseudo-positive measures. In the present article we shall
present a Markov-type error estimate for the polyharmonic cubature formula and apply this estimate to functions f
which possess an analytic extension on the ball in Cd with center 0 and sufficiently large radius. For an error estimate
of polyharmonic cubature formula based on complex methods we refer to [16]. As general background information we
mention as well our unpublished manuscript [18] which contains also instructive examples.
The paper is organized in the following way: in Section 2 we shall provide background material about spherical
harmonics and Fourier-Laplace series which is necessary for the case d > 2. In Section 3 we give a short review of properties
of the polyharmonic cubature formulas. Section 4 contains the main result of the paper – an error estimate for T (s) which
is based on the error estimate of Markov for quadratures.
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
2
65
Polyharmonic polynomials and Spherical harmonics
€ Š
We shall write x ∈ Rd in spherical coordinates x = rθ with θ ∈ Sd−1 . Let Hk Rd be the set of all harmonic homogeneous
€ Š
complex-valued polynomials of degree k. Then f ∈ Hk Rd is called a solid harmonic and the restriction of f to Sd−1 a
spherical harmonic of degree k and we set
€ Š
ak := dim Hk Rd ,
(10)
see [28], [25], [1], [13] for details. Throughout the paper we shall assume
Yk,` : Rd → R, ` = 1, ..., ak ,
€ Š
is an orthonormal basis of Hk Rd with respect to the scalar product
Z
f , g Sd−1 :=
f (θ ) g (θ )dθ .
(11)
Sd−1
We shall often use the trivial identity Yk,` (x) = r Yk` (θ ) for x = rθ . Further we define the surface area ωd by
Z
k
ωd =
1dθ .
Sd−1
The Fourier-Laplace series of the continuous function f : BR → C, is defined by the formal expansion
f (rθ ) =
ak
∞ X
X
f k,` (r) Yk,` (θ )
(12)
k=0 `=1
where ak is defined in (10) and the Fourier-Laplace coefficient f k,` (r) is defined by
Z
f k,` (r) =
f (rθ ) Yk,` (θ ) dθ
(13)
Sd−1
for any non-negative real number r with 0 ≤ r < R.
There is a strong interplay between algebraic and analytic properties of the function
f and those of the Fourier-Laplace
coefficients f k,` . For example, if f (x) is a polynomial in the variable x = x 1 , ..., x d then the Fourier-Laplace coefficient f k,`
is of the form f k,` (r) = r k pk,` r 2 where pk,` is a univariate polynomial, see e.g. in [28] or [26]. Hence, the Fourier-Laplace
series (12) of a polynomial f (x) is equal to
f (x) =
ak
deg
Xf X
pk,` (|x|2 )Yk,` (x)
(14)
k=0 `=1
where deg f is the total degree of f and pk,` is a univariate polynomial of degree ≤ deg f − k. This representation is often
called the Gauss representation.
A similar formula is valid for a much larger class of functions. Let us recall that a function f : G → C defined on an
open set G in Rd is called polyharmonic of order N if f is 2N times continuously differentiable and
∆N u (x) = 0
for all x ∈ G where
∆=
∂2
∂
x 12
+ ... +
(15)
∂2
∂ x d2
is the Laplace operator and ∆N the
¦ N -th iterate of ©∆. The theorem of Almansi states that for a polyharmonic function f of
order N defined on the ball BR = x ∈ Rd : |x| < R there exist univariate polynomials pk,` (t) of degree ≤ N − 1 such that
f (x) =
ak
∞ X
X
pk,` (|x|2 )Yk,` (x)
(16)
k=0 `=1
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
66
where convergence of the sum is uniform on compact subsets of BR , see e.g. [26], [3], [2] and [17] for further extensions.
Neglecting at the moment questions of convergence we see that
Z
ak Z
∞ X
X
f (x) dµ (x) =
pk,` (|x|2 )Yk,` (x) dµ (x) .
k=0 `=1
Note that pk,` is a univariate function depending on |x|2 and note that |x|2s Yk,` (x) is indeed a polynomial and therefore
Z
|x|2s Yk,` (x) dµ (x)
can be expressed as a sum of monomial moments. The above mentioned Gauss decomposition just says that each
multivariate polynomial f (x) is indeed a linear combination of polynomials of the type |x|2s Yk,` (x) .
These considerations have led us to the following definition: a signed measure µ with support in BR ⊂ Rd is pseudopositive with respect to the orthonormal basis Yk,` , ` = 1, ..., ak , k ∈ N0 if the inequality
Z
h (|x|) Yk,` (x) dµ (x) ≥ 0
(17)
Rd
holds for every non-negative continuous function h : [0, R] → [0, ∞) and for all k ∈ N0 , ` = 1, 2, ..., ak . Then the following
can be proved, see [15].
Theorem 2.1. Let µ be a pseudo-positive measure on Rd with support in BR ⊂ Rd . Then there exist unique non-negative
measures µk,` with support in [0, R], which we call component measures, such that
Z∞
Z
h (t) dµk,` (t) =
holds for all h ∈ C [0, R]. Further
Z
f (x) dµ =
Rd
h (|x|) Yk,` (x) dµ
(18)
f k,` (r) r −k dµk,` (r)
(19)
Rd
0
ak
∞ X
X
∞
Z
k=0 `=1
0
€ Š
for each f ∈ C Rd whose Fourier-Laplace series has only finitely many non-zero terms.
ψ
Let ψ : [0, ∞) → [0, ∞) be the transformation ψ (t) = t 2 and let µk,` be the image measure of µk,` under ψ. Then (19)
becomes
Z
ak Z ∞
∞ X
X
€p Š 1
ψ
f (x) dµ =
f k,`
(20)
t t − 2 k dµk,` (t) .
Rd
k=0 `=1
0
ψ
The main idea is simple and consists in replacing in formula (20) the non-negative univariate measures µk,` by their
(s)
univariate Gauß-Jacobi quadratures νk,` of order 2s−1. Then we obtain a functional T (s) defined on the set C x 1 , x 2 , ..., x d
of all polynomials by setting
ak Z ∞
∞ X
X
€p Š 1
(s)
(s)
T
f :=
f k,`
t t − 2 k dνk,` (t) .
(21)
k=0 `=1
0
Since f is a polynomial the series is finite and therefore T (s) is well-defined.
p
Sometimes it is useful to rewrite the definition of T (s) f using the variable r instead of t. If we define ψ−1 (t) = t
(s)
(s)
(so ψ−1 is the inverse function of ψ) and if σk,` is the image measure of νk,` under ψ−1 , then we may write
a
∞ X
k
X
T (s) f =
k=0 `=1
Dolomites Research Notes on Approximation
Z
∞
(s)
f k,` (r) r −k dσk,` (r) .
0
ISSN 2035-6803
Render · Kounchev
3
67
Basic properties of the polyharmonic cubature
We shall recall from [15] and [18] some basic properties for the polyharmonic cubature formula:
Theorem 3.1. Let µ be a pseudo-positive measure with support in the ball BR . Then the functional T (s) : C x 1 , x 2 , ..., x d → C
is continuous with respect to the supremum norm provided that the summability assumption
ak Z ∞
∞ X
X
r −k dµk,` (r) < ∞
(22)
k=0 `=1
0
holds.
Proof. Since µ has support in BR the measures µk,` have support in [0, R] . For the Fourier-Laplace coefficient f k,` we have
f (r) ≤ C max f (x) for 0 ≤ r ≤ R.
k,`
|x|≤R
Hence
and
Z
Z
∞
∞
(s)
(s)
−k
f k,` (r) r dσk,` (r) ≤ C max f (x)
r −k dσk,` (r)
|x|≤R
0
0
ak
∞ X
X
T (s) f ≤ C max f (x)
|x|≤R
For the convergence in (23) it suffices to prove
Z∞
k=0 `=1
(s)
r −k dσk,` (r) ≤
0
Z
Z
∞
(s)
r −k dσk,` (r) .
(23)
0
∞
r −k dµk,` (r) .
(24)
0
This inequality follows from the extremal property of the Gauß–Jacobi quadrature, see Theorem 4.1 in Chapter 4 of
[21].
By the Riesz representation theorem there exists a signed measure σ(s) with support in the closed ball BR such that
Z
T (s) f =
f (x) dσ(s) (x)
BR
for all continuous functions f : BR → C. Moreover, the component measures of the pseudo–positive measure σ(s) are
(s)
exactly the univariate measures σk,` .
Note that the summability condition (22) can be rephrased in terms of the measure µ by the identity
Z∞
Z
x
−k
dµ.
r dµk,` (r) =
Yk,`
|x|
0
Rd
We summarize the results in the following
Theorem 3.2. Let µ be a pseudo-positive signed measure with support in the closed ball BR satisfying the summability condition
(22). Then for each natural number s there exists a unique pseudo-positive, signed measure σ(s) with support in BR such that
(s)
(i) The support of each component measure σk,` of σ(s) has cardinality ≤ s.
R
R
(s)
(ii) P dµ = P dσ for all polynomials P with ∆2s P = 0.
(s)
Proof. The exactness of the Gauß-Jacobi quadratures νk,` for polynomials of degree ≤ 2s − 1 implies that T (s) and µ
2s
coincide on the set of all polynomials P such that
∆ P = 0. This is due to the fact that in the Laplace–Fourier expansion
the coefficients are given by f k,` (r) = r k pk,` r 2 where pk,` are polynomials of degree 2s − 1.
Definition 3.1. The measure σ(s) constructed in the last Theorem will be called the polyharmonic Gauß-Jacobi measure
of order s for the measure µ.
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
68
The following is an analog to the theorem of Stieltjes about the convergence of the univariate Gauß–Jacobi quadrature
formulas.
Theorem 3.3. Let σ(s) be the polyharmonic Gauß-Jacobi measure of order s for the measure µ, obtained in Theorem 3.2. Then
Z
Z
f (x) dσ(s) →
f (x) dµ
for s → ∞
holds for every function f ∈ C BR .
Proof. For any polynomial P the convergence T (s) (P) → P holds for s −→ ∞. By standard results, the convergence
T (s) f → f carries over to all continuous functions f : BR → C provided there exists a constant C > 0 such that
T (s) f ≤ C max f (x) .
|x|≤R
for all natural numbers s and all f ∈ C BR .
In a similar way one can prove the following result:
Theorem 3.4. Let µ be a pseudo-positive signed measure with support
€ in ŠBR satisfying the summability condition (22) and let
σ(s) be the polyharmonic Gauß-Jacobi measure of order s. If f ∈ C 2s Rd has the property that
d 2s h
dt
for all t ∈ 0, R
2
2s
f k,`
€p Š 1 i
t t − 2 k ≥ 0,
and for all k ∈ N0 , ` = 1, 2, ..., ak , then the following inequality
Z
Z
f (x) dσ(s) ≤
f (x) dµ
holds.
Let us note that every signed measure dµ with bounded variation may be represented (non-uniquely) as a difference of
two pseudo-positive measures. We refer to [15] for instructive examples of pseudo-positive measures.
4
Error estimate of the Polyharmonic Gauss-Jacobi Cubature formula
The topic of estimation of quadrature formulas for smooth and analytic functions is a widely studied one. Beyond the
classical monographs [22], [8, p. 344], [9], we provide further and more recent publications, as [5], [10], [11], [12],
[20], [23].
We recall here the following error estimate of Markov:
Theorem 4.1. (Markov) Let ν be a non-negative measure over the interval [a, b] and let ν s be the Gauss-Jacobi measure of
order s. Define for every g ∈ C [a, b] the error
Z b
Z b
Es g :=
g (t) dν (t) −
g (t) dν (s) (t) .
a
a
If g ∈ C [a, b] then
2s
E
s
g ≤
1
sup g (2s) (ξ)
(2s)! a<ξ<b
b
Z
|Qs (t)|2 dν (t)
a
where Qs (t) is the orthogonal polynomial of degree s, with leading coefficient 1, relative to v.
We shall prove now the following analogue:
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
69
Theorem 4.2. Let 0 < R < ∞ and let ψ : [0, ∞) → [0, ∞) be defined by ψ (t) = t 2 . Let µ be a pseudo-positive signed measure
(s)
with support in BR satisfying
the summability condition (22), and let σ be the polyharmonic Gauß-Jacobi measure of order s.
Define for every f ∈ C BR the error functional
Z
Z
Es f := f (x) dµ (x) − f (x) dσ(s) (x) .
€ Š
If f ∈ C 2s BR ∩ C BR then the error Es f is less than or equal to
1
(2s)!
ak
∞ X
X
Z
2s h
€p Š 1 i
d
−2k
sup 2s f k,`
t t
(ξ)
2 dt
k=0 `=1 0<ξ<R
0
R2
2
s
ψ
Q k,` (t) dµk` .
ψ
Qsk,` (t)
Here
is the orthogonal polynomial of degree s with respect to the measure µk` , having a leading coefficient equal to 1; if
the support of µk,` has less than s points, Qsk,` is defined to be 0.
€ Š
Proof. Since f ∈ C 2s BR ∩ C BR it is easy to see that the Fourier-Laplace coefficients f k,` ∈ C 2s (0, R) ∩ C [0, R]. Let µk,`
and σk,` , k ∈ N0 , ` = 1, ..., ak , and σ(s) be as in Theorem 3.2. From the definitions it follows
ZR
ak Z R
∞ X
X
(s)
Es f =
f k,` (r) r −k dµk,` −
f k,` (r) r −k dσk,` .
k=0 `=1
0
0
Further f k,` (r) r −k is integrable with respect to µk,` since f k,` is continuous on [0, R] and condition (22) holds. Let us fix
(s)
the pair of indices (k, `) . If the support of µk,` has less than s points we know that µk,` = σk,` . So assume that the support
ψ
(s)
of µk,` has at least s points. Then the support of µk,` has at least s points and in our construction νk,` is the Gauß-Jacobi
measure of
ψ
µk,` .
Consequently
€
R
Z
Š
f k,` (r) r
e f k,` :=
−k
dµk,` (r) −
0
=
Z
Z
R
(s)
f k,` (r) r −k dσk,` (r)
0
R2
f k,`
€p Š 1
ψ
t t − 2 k dµk,` (t) −
Z
R2
f k,`
€p Š 1
(s)
t t − 2 k dνk,` (t) .
0
0
€p Š 1
By Markov’s error estimate one obtains with g k,` (t) := f k,`
t t − 2 k the inequality
€ Š
e f k,` ≤
1
Z
(2s)
sup g k,` (ξ)
(2s)! 0<ξ<R2
R2
02
2
s
ψ
Q k,` (t) dµk,` (t) .
The proof is complete.
Now we are going to apply the results for holomorphic functions in several variables. We define the complex ball in Cd
with center 0 and radius τ by
d
X
2
w < τ2 }.
BτC = { w1 , ..., w d ∈ Cd :
j
j=1
We assume that f is holomorphic on
BτC
for τ > R. For fixed θ ∈ Sd−1 we define a map
ϕθ : {z ∈ C : |z| < τ} → BτC by ϕθ (z) = zθ
which is clearly holomorphic. Hence fθ defined by fθ (z) = f (zθ ) = f ◦ ϕθ (z) is holomorphic. It follows that f k,` (z)
defined by
Z
f k,` (z) =
f (zθ ) Yk,` (θ ) dθ
(25)
Sd−1
is a holomorphic extension of f k,` to {z ∈ C : |z| < τ}. For further material about analytic extensions of Fourier-Laplace
series and Fourier-Laplace coefficients we refer to [14], [17] and [24].
Now we need the following result:
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
70
Lemma 4.3. Let f be a holomorphic function on the open ball BτC for τ > 0. Let f k,` be the Fourier-Laplace coefficient of f and
define
€p Š
pk,` (t) = f k,`
t · t −k/2
for 0 < t < τ2 Then the following inequality
s
€
Š ρ 2−k s!
d
p
iu
≤ ω
max
f
e
ρθ
p
(t)
s+1
d
k,`
d ts
u∈[0,2π],θ ∈Sd−1
ρ2 − t
(26)
holds for all 0 < t < ρ 2 < τ2 and for all natural numbers s.
Proof. We apply Cauchy-Schwarz inequality to the integral (25) obtaining
Z
Z
f (z)2 ≤
f (zθ )2 dθ ·
Y (θ )2 dθ .
k,`
k,`
Sd−1
Sd−1
Since Yk,` is orthonormal we obtain for z = |z| e and |z| = ρ
iu
f
2
k,` (z) ≤ ωd
max
u∈[0,2π],θ ∈Sd−1
€
Š2
f e iu ρθ .
(27)
Let us recall the Cauchy estimates for a holomorphic function g in the ball |z| < τ applied for |z| = ρ
n!
g (n) (0) ≤
max g (z)
ρ n |z|=ρ
We apply this estimate to the holomorphic function f k,` (z) and n = m + k and we use (27):
(k + m)!
€
d m+k
Š
p
iu
ω
max
f
e
ρθ
.
m+k f k,` (0) ≤
d
dz
ρ m+k
u∈[0,2π],θ ∈Sd−1
(28)
Since f k,` (z) is holomorphic for |z| < τ we can write f k,` as a power series. Further it is known (see [6]) that
( j)
f k,` (0) = 0 for j = 0, ..., k − 1,
Hence we can write for |z| < τ
f k,` (z) =
∞
X
1 dm
m=k
m! d r m
f k,` (0) · z m .
It is known that r −k f k,` (r) is an even function (see [6]), hence we can obtain a description for the function pk,` r 2 :
∞
X
€ Š
pk,` r 2 = r −k f k,` (r) =
m=0
Then for t = r 2 we conclude that
pk,` (t) =
We infer that
ds
∞
X
dt
m=s
p (t) =
s k,`
∞
X
1
d 2m+k
1
(k + 2m)! d r 2m+k
d 2m+k
(k + 2m)! d r 2m+k
m=0
1
m!
f k,` (0) · r 2m .
f k,` (0) · t m .
d 2m+k
(k + 2m)! (m − s)! d r 2m+k
f k,` (0) · t (m−s) .
Now (28) implies
s
m−s
∞
€
Š 1 X
d
p
m!
t
iu
≤ ω
p
(t)
max
f
e
ρθ
.
d
d t s k,` k+2s
d−1
(m − s)! ρ 2
ρ
u∈[0,2π],θ ∈S
m=s
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
For |x| < 1 we have
and we see that
∞
X
m!
m=s
(m − s)!
x m−s =
∞
ds X
dx
s
xm =
m=0
ds
1
d ts 1 − x
71
= s! (1 − x)−s−1
s
€
Š s!
p
d
t −s−1
iu
≤ ω
p
(t)
max
f
e
ρθ
1
−
k,`
d
d ts
ρ2
ρ k+2s
u∈[0,2π],θ ∈Sd−1
which gives (26).
Combining the last two results we obtain:
Theorem 4.4. Let µ be a pseudo-positive signed measure with support in BR satisfying
the summability condition (22) and let
σ(s) be the polyharmonic Gauß-Jacobi measure of order s. Then the error Es f is less than or equal to
p
ωd ρ 2
2s+1
ρ 2 − R2
ak
∞ X
X
1
f (w)
max
k
w∈Cn ,|w|≤ρ
ρ
k=0 `=1
Z
0
R2
2
s
ψ
Q k,` (t) dµk,` (t)
for all functions f : BR → C which possess a holomorphic extension to the complex ball BτC for τ > R where ρ is any number
with R < ρ < τ.
Proof. By Theorem 4.2 the error Es f is less than or equal
ak
∞ X
X
1
(2s)!
2s h
Z
€p Š 1 i
d
sup 2s f k,`
t t − 2 k (ξ)
2 dt
k=0 `=1 0<ξ<R
R2
2
s
ψ
Q k,` (t) dµk` .
0
€p Š 1
t t − 2 k shows that
Lemma 4.3 applied for index 2s and for pk,` (t) = f k,`
2s
€
Š ρ 2−k (2s)!
p
d
iu
≤ ω
p
(t)
max
f
e
ρθ
2s+1 .
k,`
d
d t 2s
u∈[0,2π],θ ∈Sd−1
ρ2 − t
Using that ρ 2 − ξ ≥ R2 for 0 < ξ < R2 we conclude that the error Es f is less than or equal
p
ωd
max
u∈[0,2π],θ ∈Sd−1
ak
∞ X
€
Š X
f e iu ρθ k=0 `=1
ρ 2−k
2s+1
ρ 2 − R2
Z
R2
0
2
s
ψ
Q k,` (t) dµk`
and the statement is proven.
We can simplify the estimate in the following way:
Theorem 4.5. Let µ be a pseudo-positive signed measure with support in BR satisfying
the summability condition (22) and let
σ(s) be the polyharmonic Gauß-Jacobi measure of order s. Then the error Es f is less than or equal to
p
ak k Z R
∞ X
X
ωd ρ 2 R2s
R
max
f
(w)
r −k dµk,` (r)
2s+1
w∈Cn ,|w|≤ρ
ρ
ρ 2 − R2
0
k=0 `=1
for all functions f : BR → C which possess a holomorphic extension to the complex ball BτC for τ > R where ρ is any number
with R < ρ < τ.
Proof. Note that the polynomial Qsk,` (t) of degree s is of the form
€
Š €
Š
Qsk,` (t) = t − t 1,k,` .... t − t s,k,`
where the points t j,k,` are in the interval 0, R2 . It follows that t − t j,k.` < R2 and we obtain the estimate
Z
0
R2
Z
2
s
ψ
Q k,` (t) dµk,` (t) ≤ R2s
Dolomites Research Notes on Approximation
0
R2
ψ
1dµk,` (t) = R2s
Z
R
1dµk,` (r) .
0
ISSN 2035-6803
Render · Kounchev
Since
Z
R
1dµk,` (r) =
0
Z
R
r k r −k dµk,` (r) ≤ Rk
0
72
Z
R
r −k dµk,` (r)
0
we can finally estimate
1
ρk
Z
0
R2
k Z R
2
R
s
ψ
2s
r −k dµk,` (r)
Q k,` (t) dµk,` (t) ≤ R
ρ
0
and in view of Theorem 4.4 the statement is proved.
Finally we see that
R2
ρ − R2
2
<1
is equivalent to the condition 2R2 < ρ 2 . Thus for functions f which have a holomorphic extension to the complex ball with
radius τ > 2R2 we obtain an estimate where the error decreases rapidly when the order of the polyharmonic cubature is
increased.
Acknowledgment
Both authors thank the Alexander von Humboldt Foundation.
References
[1] G.E. Andrews, R. Askey, R. Roy, Special functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press,
Cambridge, 1999.
[2] N. Aronszajn, T.M. Creese, L.J. Lipkin, Polyharmonic Functions, Clarendon Press, Oxford 1983.
[3] V. Avanissian, Cellule d’harmonicité et prolongement analytique complexe, Hermann, Paris, 1985.
[4] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, second edition, Springer, New York, 2001.
[5] N. S. Bakhvalov, On the optimal speed of integrating analytic functions, U.S.S.R. Comput. Math. Math. Phys.7 (1967), 63–75.
[6] M.S. Baouendi, C. Goulaouic, L.J. Lipkin, On the operator ∆r 2 + µ (∂ /∂ r) r + λ, J. Differential Equations 15 (1974), 499–509.
[7] R. Cools, An Encyclopaedia of Cubature Formulas, J. Complexity, 19 (2003), 445–453.
[8] P. Davis, Interpolation and Approximation. Dover Publications Inc., New York, 1975.
[9] P. Davis, P. Rabinowitz, Methods of Numerical Integration. Second edition. Computer Science and Applied Mathematics. Academic
Press, Inc., Orlando, FL, 1984.
[10] W. Gautschi, R.S.Varga, Error Bounds for Gaussian Quadrature of Analytic Functions, SIAM J. Numer. Anal. 20, p. 1170-1186 (1983).
[11] M. Goetz, Optimal quadrature for analytic functions, J. Comput. Appl. Math. 137 (2001), 123–133.
[12] M. Kzaz, Convergence acceleration of some Gaussian quadrature formulas for analytic functions, J. Appl. Numer. Math. 10 (1992),
481–496.
[13] O. Kounchev, Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, San Diego, 2001.
[14] O. Kounchev, H. Render, Holomorphic Continuation via Fourier-Laplace series, Contemporary Mathematics 455 (2008), 197–205.
[15] O. Kounchev, H. Render, A moment problem for pseudo-positive definite functionals, Arkiv fœr Matematik, 48 (2010), 97-120.
[16] O. Kounchev, H. Render, Polyharmonic Hardy spaces on the Complexified Annulus and Error estimates of Cubature formulas, Results
Math. 62 (2012), 377–403.
[17] O. Kounchev, H. Render, Polyharmonic functions of infinite order on annular regions, to appear in Tohoku Math. Journal.
[18] O. Kounchev, H. Render, Reconsideration of the multivariate moment problem and a new method for approximating multivariate
integrals, electronic version at arXiv:math/0509380v1 [math.FA]
[19] O. Kounchev, H. Render, Multivariate Moment Problem, Hardy Spaces, and Orthogonality, in preparation.
[20] M. A. Kowalski, A. G. Werschulz, and H. Wozniakowski, Is Gauss quadrature optimal for analytic functions?, Numer. Math. 47 (1985),
89-98.
[21] M. Krein, A. Nudelman, The Markov moment problem and extremal problems, Amer. Math. Soc., Providence, R.I., 1977.
[22] V. Krylov, Approximate calculation of integrals. Translated by Arthur H. Stroud, The Macmillan Co., New York-London, 1962.
Dolomites Research Notes on Approximation
ISSN 2035-6803
Render · Kounchev
73
[23] G. Milovanovic, M.M. Spalevic, Error bounds for Gauss-Turán quadrature formulas of analytic functions, Math. Comp. 72 (2003),
1855–1872.
[24] M. Morimoto, Analytic Functionals on the Sphere, Translation of Mathematical Monographs, Vo. 178, Amer. Math. Soc., Providence,
Rhode Island 1998.
[25] R. Seeley, Spherical harmonics, Amer. Math. Monthly, 73 (1966), 115–121.
[26] S.L. Sobolev, Cubature formulas and modern analysis. An introduction. Translated from the 1988 Russian edition. Gordon and Breach
Science Publishers, Montreux, 1992.
[27] S. Sobolev, V. Vaskevich, The theory of cubature formulas, Springer, Berlin, 1997.
[28] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton University Press, 1971.
[29] A.H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, Englewood Cliffs, N.J., 1971.
[30] A.H. Stroud, Numerical quadrature and solution of ordinary differential equations. Springer-Verlag, New York-Heidelberg, 1974.
[31] A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Dover Publications, 1990.
[32] I. Vekua, New Methods for Solving Elliptic Equations. Wiley, New York, 1967.
Dolomites Research Notes on Approximation
ISSN 2035-6803