# Convolution structures on discrete spaces

Selvanathan, Saroja
(1985)
*Convolution structures on discrete spaces.*
Masters by Research thesis, Murdoch University.

## Abstract

The study of hypergroups in harmonic analysis was put on a firm footing with the (independent) papers published by the three authors Dunkl, Spector and Jewett in 1972. Since then work on hypergroups. together with the development of probability theory on these spaces, has progressed considerably.

This thesis deals mainly with commutative discrete hypergroup structures; in particular, the hypergroup structures on the set N of natural numbers. Chapter One contains an introduction to the thesis and some general properties of a commutative discrete hypergroup arising from its structural definition and using known results proved by other authors.

The idea of generalizing convolution structures has in fact been investigated by many authors. In 1974, Schwartz gave an axiomatic structure that leads to a generalized convolution for probability measures defined on N, via certain families of orthogonal functions. In the second chapter, we investigate how the convolution spaces of Schwartz relate to those developed by the other authors (in particular Gilewski and Urbanik) and develop basic probability theory for various Banach convolution algebras of probability measures on N.

In Chapter Three we prove that every hermitian hypergroup structure on N can be generated (in the sense of Schwartz) by a certain family of real valued continuous functions defined on a compact interval, and we characterize such structures when the generating functions are polynomials.

The fourth chapter investigates the hypergroup structures of the duals of compact groups and their hypergroup duals. From a suitable family of functions on a compact space, we develop in Chapter Five a generalized version of the results in Chapter Three to include all commutative discrete hypergroup structures.

In the final chapter we develop some basic probability theory for hypergroups.

Item Type: | Thesis (Masters by Research) |
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Murdoch Affiliation(s): | School of Mathematical and Physical Sciences |

Notes: | Note to the author: If you would like to make your thesis openly available on Murdoch University Library's Research Repository, please contact: repository@murdoch.edu.au. Thank you. |

Supervisor(s): | UNSPECIFIED |

URI: | http://researchrepository.murdoch.edu.au/id/eprint/51539 |

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