Generalized Fourier Transforms And Their Applications
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Integral Transforms of Generalized Functions and Their Applications
For those who have a background in advanced calculus, elementary topology and functional analysis - from applied mathematicians and engineers to physicists - researchers and graduate students alike - this work provides a comprehensive analysis of the many important integral transforms and renders particular attention to all of the technical aspects of the subject. The author presents the last two decades of research and includes important results from other works.
Time Frequency Analysis of Some Generalized Fourier Transforms
Author: Mohammad Younus Bhat
language: en
Publisher: BoD – Books on Demand
Release Date: 2023-09-13
In the world of physical science, important physical quantities like sound, pressure, electrical current, voltage, and electromagnetic fields vary with time. Such quantities are labeled as signals/waveforms and include oral signals, optical signals, acoustic signals, biomedical signals, radar, and sonar. Time-frequency analysis is a vital aid in signal analysis, which is concerned with how the frequency of a function (or signal) behaves in time, and it has evolved into a widely recognized applied discipline of signal processing. This book discusses the Fourier transform (FT), which is one of the most valuable and widely used integral transforms that converts a signal from time versus amplitude to frequency versus amplitude. It is one of the oldest tools in the time-frequency analysis of signals. The book includes five chapters that discuss general Fourier transforms as well as new and novel transforms such as hybrid transforms, quadratic-phase Fourier transforms, fractional Fourier transforms, linear canonical transforms, and more.
Generalized Fourier Transforms and Their Applications
This thesis centers around a generalization of the classical discrete Fourier transform. We first present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group or semisimple algebra. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms [65], we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of general semisimple algebras and quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. We give explicit counting results to find the dimension of these configuration spaces, and thus the complexity of the associated Fourier transform. Our methods give improved upper bounds for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group and compact Lie groups. We extend these results further to semisimple algebras, giving the first results for non-trivial upper bounds for computing Fourier transforms on the Brauer and Birman-Murakami-Wenzl (BMW) algebras. The extension of our algorithm to Fourier transforms on semisimple algebras is motivated by emerging applications of such transforms. In particular, Fourier transforms on the Iwahori-Hecke algebras have been used to analyze Metropolis-based systematic scanning strategies for generating Coxeter group elements [25]. We consider the Metropolis algorithm in the context of the Brauer and BMW monoids and provide systematic scanning strategies for generating monoid elements. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We translate these walks into left multiplication operators in the corresponding BMW algebra. Taking this algebraic perspective enables the use of tools from representation theory to analyze the walks; in particular, we develop a norm arising from a trace function on the BMW algebra to analyze the time to stationarity of the walks.