Orthonormal expansions for translation-invariant kernels

Filip Tronarp, Toni Karvonen

Abstract: We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ℒ2(R)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">ℒ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mrow is="true"><mo is="true">(</mo><mi mathvariant="double-struck" is="true">R</mi><mo is="true">)</mo></mrow></mrow></math>. 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.