Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

TL;DR

This paper introduces new classes of positive definite kernels on various non-standard spaces, including Hilbert, Banach, and measure spaces, with properties like strict positive definiteness and characteristicness, expanding kernel methods' applicability.

Contribution

It develops broad classes of kernels on Banach and metric spaces of negative type, providing explicit examples on $L^p$ spaces and measure sets, advancing kernel theory in non-Hilbert spaces.

Findings

New classes of kernels on non-standard spaces are characterized.

Explicit kernels on $L^p$ spaces are constructed.

The kernels are shown to be integrally strictly positive definite and characteristic.

Abstract

We present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes of kernels on Banach spaces and on metric spaces of strong negative type. The general results are used to give explicit classes of kernels on separable $L^p$ spaces and on sets of measures.