Radial positive definite functions and Schoenberg matrices with negative eigenvalues

Abstract

The main object under consideration is a class ${\Phi }_n\backslash {\Phi }_{n+1}$ of radial positive definite functions on ${\mathbb{R}}^n$ which do not admit \emph{radial positive definite continuation} on ${\mathbb{R}}^{n+1}$. We find certain necessary and sufficient conditions for the Schoenberg representation measure ${\nu }_n$ of $f\in {\Phi }_n$ in order that the inclusion $f\in {\Phi }_{n+k}$, $k\in \mathbb{N}$, holds. We show that the class ${\Phi }_n\backslash {\Phi }_{n+k}$ is rich enough by giving a number of examples. In particular, we give a direct proof of ${\Omega }_n\in {\Phi }_n\backslash {\Phi }_{n+1}$, which avoids Schoenberg's theorem, ${\Omega }_n$ is the Schoenberg kernel. We show that ${\Omega }_n(a\cdot ){\Omega }_n(b\cdot )\in {\Phi }_n\backslash {\Phi }_{n+1}$, for $a\ne b$. Moreover, for the square of this function we prove surprisingly much stronger result: ${\Omega }_n^2(a\cdot )\in {\Phi }_{2n-1}\backslash {\Phi }_{2n}$. We also show that any $f\in {\Phi }_n\backslash {\Phi }_{n+1}$, $n\ge 2$, has infinitely many negative squares. The latter means that for an arbitrary positive integer $N$ there is a finite Schoenberg matrix $\kS_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X:=\{x_j{\}}_{j=1}^m\subset {\mathbb{R}}^{n+1}$, which has at least $N$ negative eigenvalues.

24 pages

Cited by in corpus (1)

• Representations of hypergeometric functions for arbitrary parameter values and their use