Applications of Fourier Transforms to Generalized Functions

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