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 ℒ2(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.