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.
Author(s): M. Rahman
Edition: 1
Publisher: WIT Press / Computational Mechanics
Year: 2011
Language: English
Pages: 193
Tags: Математика;Функциональный анализ;
Cover......Page 1
Applications of Fourier Transforms to Generalized Functions......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
Acknowledgements......Page 16
1.2 Introductory remarks on Fourier series......Page 18
1.3 Half-range Fourier series......Page 22
1.4 Construction of an odd periodic function......Page 25
1.5 Theoretical development of Fourier transforms......Page 26
1.6 Half-range Fourier sine and cosine integrals......Page 28
1.7 Introduction to the first generalized functions......Page 30
1.8 Heaviside unit step function and its relation with Dirac's delta function......Page 33
1.9 Exercises......Page 35
References......Page 36
2.2 Definitions of good functions and fairly good functions......Page 38
2.3 Generalized functions. The delta function and its derivatives......Page 43
2.4 Ordinary functions as generalized functions......Page 51
2.5 Equality of a generalized function and an ordinary function in an interval......Page 53
2.6 Simple definition of even and odd generalized functions......Page 54
2.7 Rigorous definition of even and odd generalized functions......Page 55
2.8 Exercises......Page 59
References......Page 61
3.2 Non-integral powers......Page 62
3.3 Non-integral powers multiplied by logarithms......Page 69
3.4 Integral powers of an algebraic function......Page 71
3.5 Integral powers multiplied by logarithms......Page 78
3.6 Summary of results of Fourier transforms......Page 81
3.7 Exercises......Page 92
References......Page 93
4.2 The Riemann–Lebesgue lemma......Page 94
4.3 Generalization of the Riemann–Lebesgue lemma......Page 96
4.4 The asymptotic expression of the Fourier transform of a function with a finite number of singularities......Page 99
4.5 Exercises......Page 119
References......Page 120
5.2 Convergence and uniqueness of a trigonometric series......Page 122
5.3 Determination of the coefficients in a trigonometric series......Page 124
5.4 Existence of Fourier series representation for any periodic generalized function......Page 127
5.5 Some practical examples: Poisson's summation formula......Page 129
5.6 Asymptotic behaviour of the coefficients in a Fourier series......Page 136
5.7 Exercises......Page 143
References......Page 144
6.1 Introduction......Page 146
6.2 Some preliminaries leading to the fast Fourier transforms......Page 147
6.3 The discrete Fourier transform......Page 160
6.4 The fast Fourier transform......Page 166
6.5 Mathematical aspects of FFT......Page 167
6.7 Cooley–Tukey algorithms......Page 169
6.8 Application of FFT to wave energy spectral density......Page 170
6.9 Exercises......Page 172
References......Page 173
A.1 Fourier transforms g(y)=F{f(x)}=∫∞-∞ f(x)e−2πixy dx......Page 176
B.3 Properties of impulse function......Page 178
B.5 δ -Function as generalized limits......Page 180
B.7 Frequency convolution......Page 181
Appendix C: Bibliography......Page 182
G......Page 186
W......Page 187
