Operational calculus and integral transforms for groups with finite propagation speed

TL;DR

This paper develops an operational calculus for integral transforms related to hypergroups with Laplace representations, enabling bounded operators on Banach spaces, with applications to wave equations and harmonic analysis.

Contribution

It introduces a new operational calculus framework for hypergroup-based integral transforms, extending functional calculus for cosine families in Banach spaces.

Findings

Established an operational calculus theorem for Sturm--Liouville hypergroups.

Demonstrated boundedness of operators derived from hypergroup characters.

Extended harmonic analysis techniques to hypergroup convolution settings.

Abstract

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on $E$. Examples include the Mellin transform and the Mehler--Fock transform. The paper uses functional calculus for the cosine family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm--Liouville hypergroups with Laplace representation as well as analogues to the Kunze--Stein phenomenon in the hypergroup convolution setting.