Applications of Fourier Transforms to Generalized Functions

This book explains how Fourier transforms can be applied to generalized functions. The generalized function is one of the important branches of mathematics and is applicable in many practical fields. Its applications to the theory of distribution and signal processing are especially important. The Fourier transform is a mathematical procedure that can be thought of as transforming a function from its time domain to the frequency domain.The book contains six chapters and three appendices. Chapter 1 deals with preliminary remarks on Fourier series from a general point of view and also contains an introduction to the first generalized function. Chapter 2 is concerned with the generalized functions and their Fourier transforms. Chapter 3 contains the Fourier transforms of particular generalized functions. The author has stated and proved 18 formulas dealing with the Fourier transforms of generalized functions, and demonstrated some important problems of practical interest. Chapter 4 deals with the asymptotic estimation of Fourier transforms. Chapter 5 is devoted to the study of Fourier series as series of generalized functions. Chapter 6 deals with the fast Fourier transforms to reduce computer time by the algorithm developed by Cooley-Tukey in 1965. Appendix A contains the extended list of Fourier transforms pairs; Appendix B illustrates the properties of impulse function; Appendix C contains an extended list of biographical references.While the book grew partly out of the author's course given to undergraduate and graduate students at Dalhousie University 1980-2008 and partly out of his years of experience teaching at universities around the world, it includes new material not included in other literature on the topic, including solutions to previously unsolved problems. It is written to be accessible to non-experts, with clear explanations of mathematical theory, graphical illustrations, and exercises at the end of every chapter.