Orthonormal Expansions for Translation-Invariant Kernels

TL;DR

This paper introduces a Fourier analytic method to derive explicit orthonormal basis expansions for translation-invariant kernels, including Matérn, Cauchy, and Gaussian kernels, facilitating their analysis and application.

Contribution

It provides a unified technique to construct explicit orthonormal expansions for key translation-invariant kernels using Fourier analysis.

Findings

Explicit expansions for Matérn kernels of all half-integer orders

Rational function expansion for the Cauchy kernel

Hermite function expansion for the Gaussian kernel

Abstract

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$. This allows us to derive explicit expansions on the real line for (i) Mat\'ern kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.