Representation of Polynomials by Linear Combinations of Radial Basis Functions

V E Maiorov

doi:10.1007/s00365-013-9183-5

Abstract

Let ${\mathcal {P}_{n}^{d}}$ denote the space of polynomials on ℝ d of total degree n. In this work, we introduce the space of polynomials ${\mathcal {Q}_{2 n}^{d}}$ such that ${\mathcal {P}_{n}^{d}}\subset {\mathcal {Q}_{2 n}^{d}}\subset\mathcal{P}_{2n}^{d}$ and which satisfy the following statement: Let h be any fixed univariate even polynomial of degree n and $\mathcal{A}$ be a finite set in ℝ d . Then every polynomial P from the space ${\mathcal {Q}_{2 n}^{d}}$ may be represented by a linear combination of radial basis functions of the form h(∥x+a∥), $a\in \mathcal{A}$ , if and only if the set $\mathcal{A}$ is a uniqueness set for the space ${\mathcal {Q}_{2 n}^{d}}$ .

Highlights

Let Rd be the real d-dimensional Euclidean space with norm x =( d i=1 xi2 )1/2 .Denote by Bd (a, r) = {x : x − a ≤ r} the Euclidean ball with center a and radius r.Let Bd = Bd (0, 1) and Sd−1 = { x = 1} be the unit ball and sphere in the space Rd, respectively.Let h(t) be a function defined on R+
Since a large class of functions may be approximated by polynomials, the problem of the representation of polynomials by shifts of radial basis polynomials is closely connected with the above problems
We introduce another subspace of polynomials, which is defined by the following

Summary

Introduction

Let A be a subset in Rd. Denote by R(h, A) the class of functions of the form h( x + a ), where a runs over the set A. Formed by all possible finite linear combinations of functions from the set R(h, A). Results about density of the spaces formed by linear combinations of shifts of fixed functions were obtained by Wiener (see Edwards [4]), Pinkus [11, 12], Schwartz [15], Agranovsky and Quinto [1], and many others. Since a large class of functions may be approximated by polynomials, the problem of the representation of polynomials by shifts of radial basis polynomials is closely connected with the above problems. Denote by Pnd the space of all polynomials from Pd of degree at most n.

Orthogonal System P of Polynomials on the Ball

The Gegenbauer polynomials

An Orthogonal System of Polynomials on the Sphere

An Orthogonal System of Polynomials on the Ball

Main Results

Moments of Radial Functions