Convolution algebras arising from Sturm-Liouville transforms and applications

Jason P Huffman,Henry E Heatherly

doi:10.1155/s0161171201010584

Abstract

A regular Sturm-Liouville eigenvalue problem gives rise to a related linear integral transform. Churchill has shown how such an integral transform yields, under certain circumstances, a generalized convolution operation. In this paper, we study the properties of convolution algebras arising in this fashion from a regular Sturm-Liouville problem. We give applications of these convolution algebras for solving certain differential and integral equations, and we outline an operational calculus for classes of such equations.

Highlights

Convolution operations serve as an interesting and fruitful link between analysis and algebra, for example, see [3, 4, 5, 7]
We begin the study of a class of commutative, associative algebras that arise from regular SturmLiouville differential equations and their associated integral transforms and convolutions
The nexus is provided by results due to Churchill in the 1950’s, but most readily referenced in [1, Chapter 10]. (Churchill’s work was strictly in the differential equations-integral transform setting, with no reference to algebras.)

Summary

Introduction

Convolution operations serve as an interesting and fruitful link between analysis and algebra, for example, see [3, 4, 5, 7]. In order to obtain a useful convolution operation on Ω we further assume, following an idea introduced by Churchill, that T satisfies a weighted kernel product convolution property: there exists a positive sequence ω(n) such that. It is well worth noting that most of the standard regular Sturm-Liouville transforms have the desired properties to yield SL-algebras, for example, the finite Fourier transforms and their modifications, (see [2, Chapter 11, Section 115]; several examples of concrete convolutions are given, as well as the weighted kernel product convolution properties from which they derive). Let ΣR be the direct sum of countably infinitely many copies of the algebra R This is the same as the set of all sequences which have finite supports.

Routine calculations show that

But because of the fact that