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.

Table of Contents:
Contents 1 Introduction Preliminary remarks;Introductory remarks on Fourier series; Half-range Fourier series; Verification of conjecture; Verification of conjecture; Verification of conjecture; Construction of an odd periodic function; Theoretical development of Fourier transforms; Half-range Fourier sine and cosine integrals; Introduction to the first generalized functions; Heaviside unit step function and its relation with Dirac's delta function; Exercises; References 2 Generalized functions and their Fourier transforms Introduction; Definitions of good functions and fairly good functions; Generalized functions. The delta function and its derivatives; Ordinary functions as generalized functions; Equality of a generalized function and an ordinary function in an interval; Simple definition of even and odd generalized functions; Rigorous definition of even and odd generalized functions; Exercises; References 3 Fourier transforms of particular generalized functions Introduction; Non-integral powers; Non-integral powers multiplied by logarithms; Integral powers of an algebraic function; Integral powers multiplied by logarithms; The Fourier transform of xn ln|x|; The Fourier transform of x-m ln|x|; The Fourier transform of x-m ln|x| sgn(x); Summary of results of Fourier transforms; Exercises; References 4 Asymptotic estimation of Fourier transforms Introduction; The Riemann-Lebesgue lemma; Generalization of the Riemann-Lebesgue lemma; The asymptotic expression of the Fourier transform of a function with a finite number of singularities; Exercises; References 5 Fourier series as series of generalized functions Introduction; Convergence and uniqueness of a trigonometric series; Determination of the coefficients in a trigonometric series; Existence of Fourier series representation for any periodic generalized function; Some practical examples: Poisson's summation formula; Asymptotic behaviour of the coefficients in a Fourier series; Exercises; References 6 The fast Fourier transform (FFT) Introduction; Some preliminaries leading to the fast Fourier transforms; The discrete Fourier transform; The fast Fourier transform; An observation of the discrete Fourier transforms; Mathematical aspects of FFT; Reviews of some works on FFT algorithms; Cooley-Tukey algorithms; Application of FFT to wave energy spectral density; Exercises; References Appendix A: Table of Fourier transforms Fourier transforms g(y)=F{f(x)}=oA - A f(x)e?2?ixydx Appendix B: Properties of impulse function (?(x)) at a glance Introduction; Impulse function definition; Properties of impulse function; Sifting property; Scaling property; Product of a ?-function by an ordinary function; Convolution property; ?-Function as generalized limits; Time convolution; Frequency convolution Appendix C: Bibliography

About the Author :
Matiur RAHMAN is a leading applied mathematician and an expert in wave mechanics and wave loads on offshore structures. He retired as a professor in applied mathematics at Dalhousie University in Canada, after a long, distinguished career as an educator. Dr. Rahman is also the Chairman of the International Conference on Advances in Fluid Mechanics. He has written a number of best-selling titles for Wessex Institute of Technology Press, including Mathematical Methods with Applications (a Choice Outstanding Academic Title), Complex Variables and Transform Calculus, Applied Differential Equations for Scientists and Engineers