On Translation Invariant Kernels and Screw Functions

TL;DR

This paper investigates the relationship between translation invariant kernels and screw functions, providing an alternative proof of Bochner's theorem on the real line, and deepening understanding of positive definite kernels and Hilbertian metrics.

Contribution

It offers a new proof of Bochner's theorem by leveraging the connection between Hilbertian metrics and positive definite kernels on the real line.

Findings

Established a link between Hilbertian metrics and positive definite kernels.

Provided an alternative proof of Bochner's theorem for translation invariant kernels.

Enhanced theoretical understanding of kernel functions on the real line.

Abstract

We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Bochner's theorem for translation invariant positive definite kernels on the real line (Rudin, 1962).