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 $\R^n$ which do not admit \emph{radial positive definite continuation} on $\R^{n+1}$. We find certain necessary and sufficient conditions for the Schoenberg representation measure $\nu_n$ of $f\in \Phi_n$ in order that the inclusion $f\in \Phi_{n+k}$, $k\in\N$, 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 $a\not=b$. 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 $f\in\Phi_n\backslash\Phi_{n+1}$, $n\ge2$, has infinitely many negative squares. The latter means that for an arbitrary positive integer $N$ there is a finite Schoenberg matrix $\kS_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X := \{x_j\}_{j=1}^m \subset\R^{n+1}$, which has at least $N$ negative eigenvalues.