Readings in Fourier Analysis on Finite Non-Abelian Groups

Preface We are convinced that the group-theoretic approach to spectral techniques and in particular Fourier analysis offers some important advantages, among which the possibility for an unique consideration of various classes of signals is probably the most important. In particular, that approach is a mean to transfer some important very useful results from classical Fourier analysis on the real line to other algebraic structures and different classes of signals, discrete and digital signals on these structures. Among different possible groups, finite non-Abelian groups have found some interesting and useful applications in different areas of science and engineering practice. The possibly most important are those in electrical engineering and physics. This monograph reviews some authors' research in the area of abstract harmonic analysis on finite non-Abelian groups. Most of the results discussed are already published in this or the restricted form or presented at conferences and published in conference proceedings. We have attempted to present them here in a consistent, but self-contained way and uniform notation, but aware of repeating well-known results from abstract harmonic analysis, except those needed for derivation, discussion and appreciation of the results presented. However, the results are accompanied , where that was necessary or appropriate, with a short discussion including the comments concerning their relationship to the existing results in the area. The aim of this monograph is, therefore, to provide a base for a further eventual study in abstract harmonic analysis on finite not necessarily Abelian groups, which should hopefully result into a further extending of the signal processing methods and techniques to signals modelled by functions on finite non-Abelian groups. The authors would be grateful for comments on these results, especially those suggesting their improvement or concrete applications in science and engineering practice. iv CONTENTS Outline Pretensions with this book were to offer a condensed and short, but rather self-contained monograph considering some basic and new concepts in Fourier analysis on finite non-Abelian groups providing in that way a mean for a further study and development of signal processing methods, system theory and related topics on these structures. In the first chapter some general comments about signals and their mathematical modells are given offering also the reason and, therefore, explanation for the restriction of the consideration to discrete case and finite non-Abelian structures. In attempting to determine the place of the concepts considered and to trace their relationship to related notions in a more …

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