Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

TL;DR

This paper establishes a connection between reproducing kernel Hilbert spaces and unitary representations of groups and groupoids, providing methods to construct kernels from representations and vice versa, with illustrative examples and potential applications.

Contribution

It introduces a novel framework linking RKHS theory with unitary groupoid representations, including explicit construction methods and examples.

Findings

Constructed positive definite kernels from groupoid representations

Reconstructed unitary representations from kernels using Moore-Aronszajn theorem

Provided detailed examples illustrating the theoretical framework

Abstract

The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand, how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore-Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions.