# Orthonormal Expansions for Translation-Invariant Kernels

arXiv: https://arxiv.org/abs/2206.08648 · HTML: https://mlsanity.com/paper/2206.08648/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 39 references that are in the map.
- First posted on arXiv 17 Jun 2022.

## Builds on (most cited first, 2 of 2)

- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Small Sample Spaces for Gaussian Processes](https://mlsanity.com/paper/2103.03169/) (2021)

## Built on by (most cited first, 3 of 3)

- [Safe exploration in reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2503.10352/) (2025)
- [Sampling from Gaussian Processes: A Tutorial and Applications in Global Sensitivity Analysis and Optimization](https://mlsanity.com/paper/2507.14746/) (2025)
- [Why not to use the Gaussian kernel](https://mlsanity.com/paper/2608.26974/) (2026)

## Abstract (arXiv)

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of mathscrL2(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.