Ebook: Applications of Fourier Transforms to Generalized Functions

The generalized function is one of the important branches of mathematics and has enormous applications in practical fields; in particular, its application to the theory of distribution and signal processing, which are essential in this computer age. Information science plays a crucial role and the Fourier transform is extremely important for deciphering obscured information. The book contains six chapters and three appendices. Chapter 1 deals with the preliminary remarks of a Fourier series from a general point of view. This chapter also contains an introduction to the first generalized function with graphical illustrations. Chapter 2 is concerned with the generalized functions and their Fourier transforms. Many elementary theorems are clearly developed and some elementary theorems are proved in a simple way. Chapter 3 contains the Fourier transforms of particular generalized functions. We have stated and proved 18 formulas dealing with the Fourier transforms of generalized functions, and some important problems of practical interest are demonstrated. Chapter 4 deals with the asymptotic estimation of Fourier transforms. Some classical examples of pure mathematical nature are demonstrated to obtain the asymptotic behaviour of Fourier transforms. A list of Fourier transforms is included. Chapter 5 is devoted to the study of Fourier series as a series of generalized functions. The Fourier coefficients are determined by using the concept of Unitary functions. Chapter 6 deals with the fast Fourier transforms to reduce computer time by the algorithm developed by Cooley-Tukey in1965. An ocean wave diffraction problem was evaluated by this fast Fourier transforms algorithm. Appendix A contains the extended list of Fourier transforms pairs, Appendix B illustrates the properties of impulse function and Appendix C contains an extended list of biographical references