Schoenberg Representations and Gramian Matrices of Mat\'ern Functions

TL;DR

This paper introduces a novel integral transform framework to represent Matérn functions and analyze their properties, establishing conditions for their translates to form Riesz sequences in function spaces.

Contribution

It provides a new integral transform approach for representing Matérn functions and proves their translates form Riesz sequences in L^2 and Sobolev spaces.

Findings

Matérn functions can be represented via Schoenberg's integrals ensuring positive definiteness

Translates of Matérn functions form Riesz sequences in relevant function spaces

Results extend to inverse multi-quadrics with similar properties

Abstract

We represent Mat\'ern functions in terms of Schoenberg's integrals which ensure the positive definiteness and prove the systems of translates of Mat\'ern functions form Riesz sequences in $L^2(\R^n)$ or Sobolev spaces. Our approach is based on a new class of integral transforms that generalize Fourier transforms for radial functions. We also consider inverse multi-quadrics and obtain similar results.