Representation of Polynomials by Linear Combinations of Radial Basis Functions

Abstract

Let Pnd denote the space of polynomials on ℝd of total degree n. In this work, we introduce the space of polynomials Q2nd such that Pnd ⊂ Q2nd ⊂ Pndand which satisfy the following statement: Let h be any fixed univariate even polynomial of degree n and A be a finite set in ℝd. Then every polynomial P from the space Q2nd may be represented by a linear combination of radial basis functions of the form h(∥x+a∥), a ∈ A, if and only if the set A is a uniqueness set for the space Q2nd. © 2013 The Author(s).