Relative Convolutions. I. Properties and Applications

TL;DR

This paper introduces the concept of relative convolution operators induced by Lie algebras, unifying various operator classes and enabling systematic harmonic analysis applications across multiple mathematical and physical domains.

Contribution

It defines relative convolutions, explores their properties, and connects them with classical convolutions, providing a framework for applications in PDO theory, analysis, and quantum physics.

Findings

Unified framework for various convolution operators

Connections established between relative and classical convolutions

Applications demonstrated in harmonic analysis, quantum theory, and complex analysis

Abstract

To study operator algebras with symmetries in a wide sense we introduce a notion of {\em relative convolution operators} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already studied (operators of multiplication, usual group convolutions, two-sided convolution etc.) and their different combinations. Basic properties of relative convolutions are given and a connection with usual convolutions is established. Presented examples show that relative convolutions provide us with a base for systematical applications of harmonic analysis to PDO theory, complex and hypercomplex analysis, coherent states, wavelet transform and quantum theory. KEYWORDS: Lie groups and algebras, convolution operator, representation theory, Heisenberg group, integral representations, Hardy space, Szeg\"o projector, Toeplitz operators, Fock space, Segal--Bargmann space, Bargmann projector, Dirac equation, Clifford analysis, coherent states, wavelet transform, quantization.