Orthonormal Expansions for Translation-Invariant Kernels

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) Matérn kernels of all half-integer orders in t...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Tronarp, Filip, Karvonen, Toni
Format: Article
Language:English
Subjects:
Invariants
Kernels
Laguerre functions
Rational functions
Online Access:Get full text
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Summary: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) Matérn 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.
ISSN:2331-8422