3 Basic properties of the polyharmonic cubature

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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 sand corresponding error estimates. Main results are Markov-type error estimates for differentiable functions and error estimates for functionsf which possess an analytic extension to a sufficiently large ball in the complex spaceC^d.

2000 AMS subject classification: 65D30, 32A35

Keywords:Numerical integration, cubature formula, Fourier-Laplace series.

1 Introduction

LetC

R^dbe the set of all continuous complex-valued functions on the euclidean spaceR^d. Acubature formula C is a linear functional onC

R^d

of the form

C f:=α1f x₁+....+αNf x_N

. (1)

The pointsx1, ...,x_Nare callednodesorknotsand the coefficientsα1, ...,αN∈R^theweights. A basic problem in numerical analysis is to approximate integrals of the form

Z

f(x)dµ(x)

for a (signed) measureµin the euclidean spaceR^dby suitable cubature formulas.

An important characteristic of a cubature formula is exactness: the functionalCisexact on a subspace UofC R^d

with respect to a measureµif

C f= Z

f(x)dµ(x) (2)

holds for allf∈U. IfU_sis the set of all polynomialsP_sof degree≤s, and the cubature is exact onU_sbut not onU_s+1, we say thatChasorder s. Exactness on the spaceP_scan be expressed by the identities

C(x^α) = Z

x^αdµ(x) for each multi-indexα= α1, ...,αd

∈N^d0with|α|:=α1+...+αd≤swherex^α=x₁^α¹...x_d^α^d. In the theory of cubature formula it is assumed that themoments

Z

x^αdµ(x)

aSchool of Mathematical Sciences, University College Dublin (Ireland)

bInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Str., 1113 Sofia (Bulgaria) and Interdisciplinary Center for

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for|α| ≤sexist 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-calledtruncated 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 measuresµwith support in theball B_R=¦

x∈R^d^:|x|<R©

(4) and continuous functionsf :B_R→C^where^Ris a positive number or∞, and

r=|x|=p

x²₁+....+x²_d is the euclidean norm ofx= x₁, ...,x_d

∈R^d. The unit sphere will be denoted by S^d⁻¹:=¦

x∈R^d:|x|=1© , and endowed with the rotation invariant measuredθ.

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

Y_0,0(x) =Y_0,0(rcost,rsint) = 1

p2π (5)

and

Y_k,1(x) =Y_k,1(rcost,rsint) = 1

pπr^kcoskt (6)

Y_k,2(x) =Y_k,2(rcost,rsint) = 1

pπr^ksinkt (7)

fork∈NwhereNdenotes the set of all natural numbers, andN0:=N∪{0}. A pointx∈R²is written asx= (rcost,rsint) whereris the radius ofxand(cost, sint)is in the unit sphere. The Fourier coefficients of a continuous functionf are defined by

f_k,`(r) = Z2π

0

f(rcost,rsint)·Y_k,`(cost, sint)d t.

TheFourier seriesof the continuous functionf :B_R→Cis defined by the formal expansion X∞

k=0 a_k

X

`=1

f_k,`(r)Y_k,`(θ) (8)

wherea₀=1 anda_k=2 fork∈N, andθ= (cost, sint). It is easy to see thatf_k,`is a continuous function iff is continuous.

Furthermore, iff is infinitely differentiable inB_Rthen the function fk,`(r)r⁻^k

iseven(and infinitely differentiable), see[6]. Finally, iff is a polynomial then f_k,`(r)r⁻^kis a univariate polynomial inr², see Section 2 for more details.

If f is sufficiently smooth then the Fourier series (8) converges absolutely and uniformly on compact subsets ofB_Rto the functionf(x)and one obtains that

Z

R²

f(x)dµ = X^∞

k=0 a_k

X

`=1

Z

R²

f_k,`(r)Y_k,`(θ)dµ(x)

= X∞ k=0

a_k

X

`=1

Z

R²

fk,`(r)r⁻^kYk,`(x)dµ(x).

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We shall now call a signed measureµwith support inB_R⊂R²pseudo-positiveif the inequality Z

R²

h(|x|)Y_k,`(x)dµ(x)≥0

holds for every non-negative continuous functionh:[0,R]→[0,∞)and for allk∈N0, and`=1, ...,a_k. By the Riesz representation theorem there exist unique non-negative measuresµk,`defined on[0,R], which we callcomponent measures, such that

Z∞ 0

h(t)dµk,`(t) = Z

R²

h(|x|)Y_k,`(x)dµ holds for allh∈C[0,R]. Using this notation we obtain

Z

R²

f(x)dµ= X∞ k=0

a_k

X

`=1

Z∞ 0

f_k,_`(r)r⁻^kdµk,`(r). 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∞

0

f_k,`(r)r^−kdµk,`(r). (9)

Thus we assume in our approach that the Fourier coefficients f_k,`(r)are known. One may use Fast-Fourier Transform to find approximations offk,`and to combine these with our approach in order to find cubature formulas only involving the function values off – 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 fk,`(r)r^−k is an even function for smoothf we shall require that the quadrature formula is exact for all polynomials of the formr^2j for j=0, ..., 2s−1 wheresa given natural number. By taking the transformationp

rthis means that the transformed quadrature formula should be exact for all polynomialt^jfor 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 measureµwe consider the component measuresµk,`(r). Letµ^ψ_k,`be the image measure ofµk,`for the transformationψ:[0,∞)→[0,∞) defined byψ(r) =r², so

Z∞ 0

f_k,`(r)r⁻^kdµk,`(r) = Z∞

0

f_k,`p t

t⁻^k/2dµ^ψ_k,`(t).

For the non-negative univariate measuresµ^ψk,`we shall use the univariate Gauß-Jacobi quadraturesνk,^(s)`of order 2s−1 as an approximation ofµ^ψ_k,`. Thepolyharmonic cubature T^(s) f

of degree sis then defined by T^(s) f

:=

X∞ k=0

a_k

X

`=1

Z∞ 0

f_k,`p t

t⁻¹²^kdνk,^(s)`(t).

The cubature formulaT^(s)will be defined at first only for polynomials: then the sum in the definition ofT^(s) f

is actually a finite sum and no convergence questions occur. The cubature formulaT^(s)has the property that

T^(s)

|x|^2jY_k,`(x)

= Z

|x|^2jY_k,`(x)dµ(x)

for allj=0, ..., 2s−1 and for allk∈N0,`=1, ...,a_k. This is equivalent to the functionalT^(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 inC^dwith 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 cased>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 forT^(s)which is based on the error estimate of Markov for quadratures.

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2 Polyharmonic polynomials and Spherical harmonics

We shall writex∈R^din spherical coordinatesx=rθwithθ∈S^d⁻¹. LetH_k

R^dbe the set of all harmonic homogeneous complex-valued polynomials of degreek. Then f ∈H_k

R^d

is called asolid harmonicand the restriction off toS^d⁻¹a spherical harmonicof degreekand we set

a_k:=dimH_k R^d

, (10)

see[28],[25],[1],[13]for details. Throughout the paper we shall assume

Y_k,`:R^d→R^,`=1, ...,a_k, (11)

is anorthonormal basisofH_k R^d

with respect to the scalar product f,g

S^d−1:= Z

S^d−1

f(θ)g(θ)dθ.

We shall often use the trivial identityY_k,`(x) =r^kY_k`(θ)forx=rθ. Further we define the surface areaωdby ωd=

Z

S^d−1

1dθ.

TheFourier-Laplace seriesof the continuous functionf:B_R→C, is defined by the formal expansion f(rθ) =

X∞ k=0

a_k

X

`=1

fk,`(r)Yk,`(θ) (12)

wherea_kis defined in (10) and theFourier-Laplace coefficient f_k,`(r)is defined by fk,`(r) =

Z

S^d−1

f(rθ)Yk,`(θ)dθ (13)

for any non-negative real numberrwith 0≤r<R.

There is a strong interplay between algebraic and analytic properties of the functionf and those of the Fourier-Laplace coefficientsf_k,`. For example, iff(x)is a polynomial in the variablex= x₁, ...,x_d

then the Fourier-Laplace coefficientf_k,`

is of the formf_k,`(r) =r^kp_k,` r²

wherep_k,`is a univariate polynomial, see e.g. in[28]or[26]. Hence, theFourier-Laplace series (12) of a polynomialf(x)is equal to

f(x) =

degf

X

k=0 a_k

X

`=1

p_k,`(|x|²)Y_k,`(x) (14)

where degf is the total degree off andp_k,`is a univariate polynomial of degree≤degf−k. This representation is often called theGauss representation.

A similar formula is valid for a much larger class of functions. Let us recall that a function f :G→Cdefined on an open setGinR^dis calledpolyharmonic of order Niff is 2Ntimes continuously differentiable and

∆^Nu(x) =0 (15)

for allx∈Gwhere

∆ = ∂²

∂x₁²+...+ ∂²

∂x²_d

is the Laplace operator and∆^N theN-th iterate of∆. The theorem of Almansi states that for a polyharmonic function f of orderNdefined on the ballB_R=¦

x∈R^d^:|x|<R©

there exist univariate polynomialsp_k,`(t)of degree≤N−1 such that f(x) =

X∞ k=0

a_k

X

`=1

p_k,`(|x|²)Y_k,`(x) (16)

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where convergence of the sum is uniform on compact subsets ofB_R, see e.g.[26],[3],[2]and[17]for further extensions.

Neglecting at the moment questions of convergence we see that Z

f(x)dµ(x) = X∞ k=0

a_k

X

`=1

Z

pk,`(|x|²)Yk,`(x)dµ(x).

Note thatp_k,`is a univariate function depending on|x|²and note that|x|^2sY_k,`(x)is indeed a polynomial and therefore Z

|x|^2sY_k,`(x)dµ(x)

can be expressed as a sum of monomial moments. The above mentioned Gauss decomposition just says that each multivariate polynomialf(x)is indeed a linear combination of polynomials of the type|x|^2sY_k,`(x).

These considerations have led us to the following definition: a signed measureµwith support inB_R⊂R^dispseudo- positive with respect to the orthonormal basis Y_k,`,`=1, ...,a_k,k∈N0if the inequality

Z

R^d

h(|x|)Yk,`(x)dµ(x)≥0 (17)

holds for every non-negative continuous functionh:[0,R]→[0,∞)and for allk∈N0,`=1, 2, ...,a_k. Then the following can be proved, see[15].

Theorem 2.1. Letµbe a pseudo-positive measure onR^dwith support in B_R⊂R^d. Then there exist unique non-negative measuresµk,`with support in[0,R], which we callcomponent measures, such that

Z∞ 0

h(t)dµk,`(t) = Z

R^d

h(|x|)Y_k,`(x)dµ (18)

holds for all h∈C[0,R]. Further Z

R^d

f(x)dµ= X∞ k=0

a_k

X

`=1

Z∞ 0

f_k,`(r)r⁻^kdµk,`(r) (19)

for each f ∈C R^d

whose Fourier-Laplace series has only finitely many non-zero terms.

Letψ:[0,∞)→[0,∞)be the transformationψ(t) =t²and letµ^ψk,`be the image measure ofµk,`underψ. Then (19) becomes

Z

R^d

f(x)dµ=X^∞

k=0 a_k

X

`=1

Z∞ 0

f_k,`p t

t⁻¹²^kdµ^ψ_k,`(t). (20) Themain ideais simple and consists in replacing in formula (20) the non-negative univariate measuresµ^ψ_k,`by their univariate Gauß-Jacobi quadraturesν_k,`^(s)of order 2s−1. Then we obtain a functionalT^(s)defined on the setC^x1,x₂, ...,x_d of all polynomials by setting

T^(s) f:=X^∞

k=0 a_k

X

`=1

Z∞ 0

f_k,`p t

t⁻¹²^kdν_k,`^(s)(t). (21) Sincef is a polynomial the series is finite and thereforeT^(s)is well-defined.

Sometimes it is useful to rewrite the definition ofT^(s) f

using the variablerinstead oft. If we defineψ⁻¹(t) =p t (soψ⁻¹is the inverse function ofψ)and ifσ^(s)_k,`is the image measure ofν_k,`^(s)underψ⁻¹, then we may write

T^(s) f

= X∞ k=0

a_k

X

`=1

Z∞ 0

f_k,`(r)r^−kdσ^(s)_k,`(r).

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3 Basic properties of the polyharmonic cubature

We shall recall from[15]and[18]some basic properties for the polyharmonic cubature formula:

Theorem 3.1. Letµbe a pseudo-positive measure with support in the ball B_R.Then the functional T^(s):C^x1,x2, ...,x_d

→C is continuous with respect to the supremum norm provided that thesummability assumption

X∞ k=0

a_k

X

`=1

Z∞ 0

r^−kdµk,`(r)<∞ (22)

holds.

Proof. Sinceµhas support inB_Rthe measuresµk,`have support in[0,R]. For the Fourier-Laplace coefficient fk,`we have

f_k,`(r)

≤Cmax

|x|≤R

f(x)

for 0≤r≤R.

Hence

Z∞ 0

f_k,`(r)r^−kdσ^(s)k,`(r)

≤Cmax

|x|≤R

f(x)

Z∞

0

r^−kdσ^(s)k,`(r) and

T^(s) f

≤Cmax

|x|≤R

f(x)

X∞ k=0

a_k

X

`=1

Z∞ 0

r⁻^kdσ^(s)k,`(r). (23)

For the convergence in (23) it suffices to prove Z∞

0

r^−kdσ^(s)k,`(r)≤ Z∞

0

r^−kdµk,`(r). (24)

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 measureσ^(s)with support in the closed ballB_Rsuch that T^(s) f

= Z

B_R

f(x)dσ^(s)(x)

for all continuous functions f :B_R→C. Moreover, the component measures of the pseudo–positive measureσ^(s) are exactly the univariate measuresσ^(s)_k,`.

Note that the summability condition (22) can be rephrased in terms of the measureµby the identity Z∞

0

r^−kdµk,`(r) = Z

R^d

Y_k,`

x

|x|

dµ.

We summarize the results in the following

Theorem 3.2. Letµbe a pseudo-positive signed measure with support in the closed ball B_Rsatisfying the summability condition (22). Then for each natural number s there exists a unique pseudo-positive, signed measureσ^(s)with support in B_Rsuch that

(i) The support of each component measureσ^(s)k,`ofσ^(s)has cardinality≤s.

(ii)R

P dµ=R

P dσ^(s)for all polynomials P with∆^2sP=0.

Proof. The exactness of the Gauß-Jacobi quadraturesν_k,`^(s) for polynomials of degree≤2s−1 implies thatT^(s)andµ coincide on the set of all polynomialsPsuch that∆^2sP=0. This is due to the fact that in the Laplace–Fourier expansion the coefficients are given byfk,`(r) =r^kpk,` r²

wherepk,`are polynomials of degree 2s−1.

Definition 3.1. The measureσ^(s)constructed in the last Theorem will be called thepolyharmonic Gauß-Jacobi measure of ordersfor the measureµ.

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The following is an analog to the theorem of Stieltjes about the convergence of the univariate Gauß–Jacobi quadrature formulas.

Theorem 3.3. Letσ^(s)be the polyharmonic Gauß-Jacobi measure of order s for the measureµ,obtained in Theorem3.2. Then Z

f(x)dσ^(s)→ Z

f(x)dµ for s→ ∞ holds for every function f ∈C B_R

.

Proof. For any polynomial Pthe convergence T^(s)(P)→ P holds fors−→ ∞. By standard results, the convergence T^(s) f

→f carries over to all continuous functionsf :B_R→Cprovided there exists a constantC>0 such that T⁽^s⁾ f

≤Cmax

|x|≤R

f(x)

. for all natural numberssand allf ∈C B_R

.

In a similar way one can prove the following result:

Theorem 3.4. Letµbe a pseudo-positive signed measure with support in B_Rsatisfying the summability condition (22) and let σ^(s)be the polyharmonic Gauß-Jacobi measure of order s. If f ∈C^2s

R^dhas the property that d^2s

d t^2s h

f_k,`p t

t⁻¹²^ki

≥0, for all t∈ 0,R²

and for all k∈N0,`=1, 2, ...,a_k, then the following inequality Z

f(x)dσ^(s)≤ Z

f(x)dµ holds.

Let us note that every signed measuredµwith 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) Letνbe a non-negative measure over the interval[a,b]and letν^sbe the Gauss-Jacobi measure of order s.Define for every g∈C[a,b]the error

E_s g :=

Z b

a

g(t)dν(t)− Zb

a

g(t)dν^(s)(t). If g∈C^2s[a,b]then

E_s g ≤ 1

(2s)! sup

a<ξ<b

g^(2s)(ξ)

Z b

a

|Q^s(t)|²dν(t) where Q^s(t)is the orthogonal polynomial of degree s, with leading coefficient1,relative to v.

We shall prove now the following analogue:

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Theorem 4.2. Let0<R<∞and letψ:[0,∞)→[0,∞)be defined byψ(t) =t².Letµbe a pseudo-positive signed measure with support in B_Rsatisfying the summability condition (22), and letσ^(s)be the polyharmonic Gauß-Jacobi measure of order s.

Define for every f ∈C B_R

the error functional E_s f:=

Z

f(x)dµ(x)− Z

f(x)dσ^(s)(x). If f ∈C^2s B_R

∩C B_R

then the error E_s f

is less than or equal to 1

(2s)! X∞

k=0 a_k

X

`=1

sup

0<ξ<R²

d^2s d t^2s

h f_k,`p

t t⁻¹²^ki

(ξ)

ZR²

0

Q^s_k,`(t)

2

dµ^ψ_k`.

Here Q^s_k,`(t)is the orthogonal polynomial of degree s with respect to the measureµ^ψ_k`,having a leading coefficient equal to1; if the support ofµk,`has less than s points, Q^s_k,`is defined to be0.

Proof. Sincef ∈C^2s B_R

∩C B_R

it is easy to see that the Fourier-Laplace coefficients f_k,`∈C^2s(0,R)∩C[0,R]. Letµk,`

andσk,`,k∈N0,`=1, ...,a_k, andσ^(s)be as in Theorem3.2. From the definitions it follows E_s f=X^∞

k=0 a_k

X

`=1

ZR

0

f_k,`(r)r⁻^kdµk,`− ZR

0

f_k,`(r)r⁻^kdσ^(s)_k,`.

Furtherf_k,`(r)r⁻^kis integrable with respect toµk,`sincef_k,`is continuous on[0,R]and condition (22) holds. Let us fix the pair of indices(k,`). If the support ofµk,`has less thanspoints we know thatµk,`=σ^(s)k,`. So assume that the support ofµk,`has at leastspoints. Then the support ofµ^ψ_k,`has at leastspoints and in our constructionν_k,`^(s)is the Gauß-Jacobi measure ofµ^ψk,`. Consequently

e f_k,_`

:= ZR

0

f_k,_`(r)r⁻^kdµk,`(r)− ZR

0

f_k,_`(r)r⁻^kdσ^(s)_k,`(r)

= ZR²

0

f_k,`p t

t⁻¹²^kdµ^ψ_k,`(t)− ZR²

0

f_k,`p t

t⁻¹²^kdν_k,`^(s)(t). By Markov’s error estimate one obtains withgk,`(t):=fk,`

pt

t⁻¹²^kthe inequality e

f_k,`

≤ 1 (2s)! sup

0<ξ<R²

g⁽_k,^2s)_` (ξ)

ZR²

0²

Q^s_k,_`(t)

2

dµ^ψk,`(t).

The proof is complete.

Now we are going to apply the results for holomorphic functions in several variables. We define the complex ball inC^d with center 0 and radiusτby

B^C_τ={ w₁, ...,w_d

∈C^d^:

d

X

j=1

w_j

2< τ²}.

We assume that f is holomorphic onB_τ^Cforτ >R. For fixedθ∈S^d−¹we define a map ϕθ:{z∈C^:|z|< τ} →B^C_τ byϕθ(z) =zθ

which is clearly holomorphic. Hence f_θ defined by f_θ(z) =f(zθ) = f◦ϕθ(z)is holomorphic. It follows that f_k,`(z) defined by

f_k,`(z) = Z

S^d−1

f(zθ)Y_k,`(θ)dθ (25)

is a holomorphic extension off_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:

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Lemma 4.3. Let f be a holomorphic function on the open ball B_τ^Cforτ >0.Let f_k,`be the Fourier-Laplace coefficient of f and define

p_k,`(t) =f_k,`p t

·t^−k/² for0<t< τ²Then the following inequality

d^s d t^sp_k,`(t)

≤pωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

ρ²⁻^ks!

ρ²−ts+1 (26)

holds for all0<t< ρ²< τ²and for all natural numbers s.

Proof. We apply Cauchy-Schwarz inequality to the integral (25) obtaining

fk,`(z)

2≤ Z

S^d−1

f(zθ)

2dθ· Z

S^d−1

Yk,`(θ)

2dθ.

SinceY_k,`is orthonormal we obtain forz=|z|e^iuand|z|=ρ

f_k,`(z)

2≤ωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

2

. (27)

Let us recall the Cauchy estimates for a holomorphic functiongin the ball|z|< τapplied for|z|=ρ

g⁽ⁿ⁾(0) ≤ n!

ρⁿmax

|z|=ρ

g(z)

We apply this estimate to the holomorphic functionfk,`(z)andn=m+kand we use (27):

d^m+k dz^m+kf_k,`(0)

≤(k+m)! ρ^m+k

pωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

. (28)

Sincef_k,`(z)is holomorphic for|z|< τwe can write f_k,`as a power series. Further it is known (see[6]) that f(^j)

k,` (0) =0 forj=0, ...,k−1, Hence we can write for|z|< τ

f_k,_`(z) =X^∞

m=k

1 m!

d^m

d r^mf_k,_`(0)·z^m.

It is known thatr⁻^kf_k,`(r)is an even function (see[6]), hence we can obtain a description for the functionp_k,` r² : pk,`

r²

=r⁻^kfk,`(r) = X∞ m=0

1 (k+2m)!

d^2m+k

d r^2m+kfk,`(0)·r^2m. Then fort=r²we conclude that

p_k,`(t) =X^∞

m=0

1 (k+2m)!

d^2m+k

d r^2m+kf_k,`(0)·t^m. We infer that

d^s

d t^sp_k,`(t) = X∞ m=s

1 (k+2m)!

m!

(m−s)!

d^2m+k

d r^2m+kf_k,`(0)·t^(m⁻^s). Now (28) implies

d^s d t^sp_k,`(t)

≤pωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

1 ρ^k+^2s

X∞ m=s

m!

(m−s)! t

ρ² m−s

.

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For|x|<1 we have

X∞ m=s

m!

(m−s)!x^m−s= d^s d x^s

X∞ m=0

x^m= d^s d t^s

1

1−x =s!(1−x)^−s−1 and we see that

d^s d t^spk,`(t)

≤p

ωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

s!

ρ^k+2s

1− t ρ²

−s−1

which gives (26).

Combining the last two results we obtain:

Theorem 4.4. Letµbe a pseudo-positive signed measure with support in B_Rsatisfying the summability condition (22) and let σ^(s)be the polyharmonic Gauß-Jacobi measure of order s. Then the error E_s f

is less than or equal to pωdρ²

ρ²−R²2s+1 max

w∈Cⁿ,|w|≤ρ

f(w)

X∞ k=0

a_k

X

`=1

1 ρ^k

ZR²

0

Q^s_k,`(t)

2

dµ^ψ_k,`(t)

for all functions f :B_R→Cwhich possess a holomorphic extension to the complex ball B_τ^Cforτ >R whereρis any number with R< ρ < τ.

Proof. By Theorem4.2the errorE_s f

is less than or equal 1

(2s)!

X∞ k=0

a_k

X

`=1

sup

0<ξ<R²

d^2s d t^2s

h f_k,`p

t t⁻¹²^ki

(ξ)

ZR²

0

Q^s_k,`(t)

2

dµ^ψ_k`. Lemma4.3applied for index 2sand forp_k,`(t) =f_k,`p

t

t⁻¹²^kshows that

d^2s d t^2sp_k,`(t)

≤pωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

ρ²⁻^k(2s)! ρ²−t2s+1. Using thatρ²−ξ≥R²for 0< ξ <R²we conclude that the errorE_s f

is less than or equal pωd max

u∈[0,2π],θ∈S^d−1

f

e^iuρθ

X∞ k=0

a_k

X

`=1

ρ²^−k ρ²−R²2s+1

ZR²

0

Q^s_k,`(t)

2

dµ^ψ_k`

and the statement is proven.

We can simplify the estimate in the following way:

Theorem 4.5. Letµbe a pseudo-positive signed measure with support in B_Rsatisfying the summability condition (22) and let σ^(s)be the polyharmonic Gauß-Jacobi measure of order s. Then the error E_s f

is less than or equal to pωdρ²R^2s

ρ²−R²2s+1 max

w∈Cⁿ,|w|≤ρ

f(w)

X∞ k=0

a_k

X

`=1

R ρ

kZR

0

r^−kdµk,`(r)

for all functions f :B_R→Cwhich possess a holomorphic extension to the complex ball B_τ^Cforτ >R whereρis any number with R< ρ < τ.

Proof. Note that the polynomialQ^s_k,`(t)of degreesis of the form Q^s_k,`(t) =

t−t_1,k,` ....

t−t_s,k,` where the pointst_j,k,`are in the interval 0,R²

. It follows that t−t_j,k.`

<R²and we obtain the estimate ZR²

0

Q^s_k,`(t)

2

dµ^ψ_k,`(t)≤R^2s ZR²

0

1dµ^ψ_k,`(t) =R^2s ZR

0

1dµk,`(r).

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Since

ZR

0

1dµk,`(r) = ZR

0

r^kr⁻^kdµk,`(r)≤R^k ZR

0

r⁻^kdµk,`(r) we can finally estimate

1 ρ^k

ZR²

0

Q^s_k,`(t)

2

dµ^ψ_k,`(t)≤R^2s R

ρ kZR

0

r^−kdµk,`(r) and in view of Theorem4.4the statement is proved.

Finally we see that

R² ρ²−R² <1

is equivalent to the condition 2R²< ρ². Thus for functionsf which have a holomorphic extension to the complex ball with radiusτ >2R²we 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.

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