Integral geometry, hypergroups, and I.M. Gelfand's question

TL;DR

This paper explores the connection between integral geometry problems and harmonic analysis on hypergroups, providing new examples and insights into why some problems relate to group harmonic analysis while others do not.

Contribution

It introduces a framework linking integral geometry problems to harmonic analysis on dual pairs of commutative hypergroups, and constructs new hypergroup examples.

Findings

Established a duality framework for hypergroups in integral geometry

Constructed new examples of hypergroups relevant to harmonic analysis

Clarified the relation between integral geometry problems and hypergroup harmonic analysis

Abstract

This note is an attempt to give an answer for the following old I.M. Gelfand's question: why some important problems of integral geometry (e.g., the Radon transform and others) are related to harmonic analysis on groups, but for other quite similar problems such relations are not clear? In the note we examine standard problems of integral geometry generating harmonic analysis (the Plancherel theorem etc.) on pairs of commutative hypergroups that are in a duality of Pontryagin's type. As a result new meaningful examples of hypergroups are constructed.