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Chapter 1 - Fourier Analysis and Fourier Transform Chapter 2 - Fourier Series Chapter 3 - Discrete Fourier Transform and Discrete-Time Fourier Transform Chapter 4 - Spherical Harmonics

Chapter 1 - Fourier Series Chapter 2 - Convergence of Fourier Series and Dirichlet Kernel Chapter 3 - Fourier Amplitude Sensitivity Testing and Gibbs Phenomenon Chapter 4 - Gamma Function Chapter 5 - Digamma Function and Factorial Chapter 6 - Incomplete Gamma Function and Lanczos Approximation Chapter 7 - Multiplication Theorem and Particular Values of the Gamma Function Chapter 8 - Polygamma Function and Stirling's Approximation

This is a concise introduction to Fourier series covering history, major themes, theorems, examples and applications. It can be used to learn the subject, and also to supplement, enhance and embellish undergraduate courses on mathematical analysis.The book begins with a brief summary of the rich history of Fourier series over three centuries. The subject is presented in a way that enables the reader to appreciate how a mathematical theory develops in stages from a practical problem (such as conduction of heat) to an abstract theory dealing with concepts such as sets, functions, infinity and convergence. The abstract theory then provides unforeseen applications in diverse areas.The author starts out with a description of the problem that led Fourier to introduce his famous series. The mathematical problems this leads to are then discussed rigorously. Examples, exercises and directions for further reading and research are provided, along with a chapter that provides materials at a more advanced level suitable for graduate students. The author demonstrates applications of the theory to a broad range of problems.The exercises of varying levels of difficulty that are scattered throughout the book will help readers test their understanding of the material.

In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived.The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

The study of sets of uniqueness for trigonometric series has a long history, originating in the work of Riemann, Heine, and Cantor in the mid-nineteenth century. Since then it has been a fertile ground for numerous investigations involving real analysis, classical and abstract harmonic analysis, measure theory, functional analysis and number theory. In this book are developed the intriguing and surprising connections that the subject has with descriptive set theory. These have only been discovered recently and the authors present here this novel theory which leads to many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. In order to make the material accessible to logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Thus the book is essentially self-contained and will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis.

This book is concerned with the modern theory of Fourier series. Treating developments since Zygmund's classic study, the authors begin with a thorough discussion of the classical one-dimensional theory from a modern perspective. The text then takes up the developments of the 1970s, beginning with Fefferman's famous disc counterexample. The culminating chapter presents Cordoba's geometric theory of Kayeka maximal functions and multipliers. Research workers in the fields of Fourier analysis and harmonic analysis will find this a valuable account of these developments. Second year graduate students, who are familiar with Lebesgue theory and are acquainted with distributions, will be able to use this as a textbook which will bring them up to the exciting open questions in the field.