Schoenberg matrices of radial positive definite functions and Riesz sequences in $L^2(\R^n)$

Abstract

Given a function $f$ on the positive half-line $\R_+$ and a sequence (finite or infinite) of points $X=\{x_k\}_{k=1}^ω$ in $\R^n$, we define and study matrices $\kS_X(f)=\|f(|x_i-x_j|)\|_{i,j=1}^ω$ called Schoenberg's matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators $S_X(f)$ on $\ell^2(\N)$. We provide conditions on $X$ and $f$ for the latter to hold. If $f$ is an $\ell^2$-positive definite function, such conditions are given in terms of the Schoenberg measure $σ(f)$. We also approach Schoenberg's matrices from the viewpoint of harmonic analysis on $\R^n$, wherein the notion of the strong $X$-positive definiteness plays a key role. In particular, we prove that \emph{each radial $\ell^2$-positive definite function is strongly $X$-positive definite} whenever $X$ is separated. We also implement a "grammization" procedure for certain positive definite Schoenberg's matrices. This leads to Riesz--Fischer and Riesz sequences (Riesz bases in their linear span) of the form $\kF_X(f)=\{f(x-x_j)\}_{x_j\in X}$ for certain radial functions $f\in L^2(\R^n)$. Examples of Schoenberg's operators with various spectral properties are presented.