Orthonormal Expansions for Translation-Invariant Kernels. (arXiv:2206.08648v1 [math.CA])

Web: http://arxiv.org/abs/2206.08648

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.

arxiv math translation