Jaen J. Approx. 5(1) (2013), 1–18
ISSN: 1889-3066
c 2013 Universidad de Jaén
Jaen Journal
on Approximation
Web site: jja.ujaen.es
Moments of shifted radial functions with
respect to orthogonal system of polynomials
on the ball
Vitaly E. Maiorov
Abstract
The moments hg, PI i of shifted radial functions g with respect to some orthogonal polynomial system {PI } of functions on the unit ball are calculated. Also with
the help of these moments we prove the following result: given a natural number n
we denote by Pnd the space of polynomials on Rd of total degree n and by Qd2n some
d
. Let Hn be the space of harmonic
space of polynomials such that Pnd ⊂ Qd2n ⊂ P2n
polynomials of degree n and Nn be the dimension of Hn . Then every polynomial p
in the space Qd2n may be represented by a linear combination of Nn shifted radial
1
, ak ∈ Rd , k = 1, ..., Nn , if and only if
functions of the form gk (kx + ak k), gk ∈ P2n
the set {a1 , ..., aNn } is a uniqueness set for the space Hn .
Keywords: radial functions, moments, approximation, harmonic analysis.
MSC: 41A05, 41A30, 65D05, 42B05.
§1.
Introduction and main results
Pd
Let Rd be the real Euclidean space with norm kxk = ( i=1 x2i )1/2 . Let s = (s1 , ..., sd )
be any vector with nonnegative integer coordinates. We denote xs = xs11 · · · xsdd and
|s| = s1 + · · · + sd . Consider the space P d = span{xs : s ∈ Zd+ } of all polynomials on
1
Communicated by
M. Buhmann
Received
December 12, 2011
Accepted
February 11, 2013