Fourier-Stieltjes transform defined by induced representation on locally compact groups

TL;DR

This paper extends the Fourier-Stieltjes transform to locally compact groups using induced representations, generalizing classical results from compact groups and establishing a series form transform with key properties.

Contribution

It introduces a new Fourier-Stieltjes transform for locally compact groups based on induced representations, extending orthogonality relations and transform properties.

Findings

Extended Fourier-Stieltjes transform to locally compact groups.

Established a series form of the transform that is linear, continuous, and invertible.

Generalized Schur orthogonality relations to this broader setting.

Abstract

In this work we extend the Fourier-Stieltjes transform of a vector measure and a continuous function defined on compact groups to locally compact groups. To do so, we consider a representation L of a normal compact subgroup K of a locally compact group G, and we use a representation of G induced by that of L. Then, we define the Fourier-Stieltjes transform of a vector measure and that of a continuous function with compact support defined on G from the representation of G. Then, we extend the Shur orthogonality relation established for compact groups to locally compact groups by using the representations of G induced by the unitary representations of one of its normal compact subgroups. This extension enables us to develop a Fourier-Stieltjes transform in series form that is linear, continuous, and invertible.