Representation of Polynomials by Linear Combinations of Radial Basis Functions

Let ${\mathcal {P}_{n}^{d}}$ denote the space of polynomials on ? ^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 ? ^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  ${\mathcal {Q}_{2 n}^{d}}$ .