Preface ........................................................ xi
Acknowledgements ............................................... xv
1 Introduction ................................................. 1
1.1 Preliminary remarks ..................................... 1
1.2 Introductory remarks on Fourier series .................. 1
1.3 Half-range Fourier series ............................... 5
1.3.1 Verification of conjecture (1.14) ................ 6
1.3.2 Verification of conjecture (1.15) ................ 6
1.3.3 Verification of conjecture (1.16) ................ 7
1.4 Construction of an odd periodic function ................ 8
1.5 Theoretical development of Fourier transforms ........... 9
1.6 Half-range Fourier sine and cosine integrals ........... 11
1.7 Introduction to the first generalized functions ........ 13
1.8 Heaviside unit step function and its relation with
Dirac's delta function ................................. 16
1.9 Exercises .............................................. 18
References .................................................. 19
2 Generalized functions and their Fourier transforms .......... 21
2.1 Introduction ........................................... 21
2.2 Definitions of good functions and fairly good
functions .............................................. 21
2.3 Generalized functions. The delta function and its
derivatives ............................................ 26
2.4 Ordinary functions as generalized functions ............ 34
2.5 Equality of a generalized function and an ordinary
function in an interval ................................ 36
2.6 Simple definition of even and odd generalized
functions .............................................. 37
2.7 Rigorous definition of even and odd generalized
functions .............................................. 38
2.8 Exercises .............................................. 42
References .................................................. 44
3 Fourier transforms of particular generalized functions ...... 45
3.1 Introduction ........................................... 45
3.2 Non-integral powers .................................... 45
3.3 Non-integral powers multiplied by logarithms ........... 52
3.4 Integral powers of an algebraic function ............... 54
3.5 Integral powers multiplied by logarithms ............... 61
3.5.1 The Fourier transform of xn ln|x| ............... 61
3.5.2 The Fourier transform of x-m ln|x| .............. 62
3.5.3 The Fourier transform of x-m ln|x| sgn(x) ....... 63
3.6 Summary of results of Fourier transforms ............... 64
3.7 Exercises .............................................. 75
References .................................................. 76
4 Asymptotic estimation of Fourier transforms ................. 77
4.1 Introduction ........................................... 77
4.2 The Riemann-Lebesgue lemma ............................. 77
4.3 Generalization of the Riemann-Lebesgue lemma ........... 79
4.4 The asymptotic expression of the Fourier transform
of a function with a finite number of singularities .... 82
4.5 Exercises ............................................. 102
References ................................................. 102
5 Fourier series as series of generalized functions .......... 105
5.1 Introduction .......................................... 105
5.2 Convergence and uniqueness of a trigonometric
series ................................................ 105
5.3 Determination of the coefficients in a
trigonometric series .................................. 107
5.4 Existence of Fourier series representation for any
periodic generalized function ......................... 110
5.5 Some practical examples: Poisson's summation
formula ............................................... 112
5.6 Asymptotic behaviour of the coefficients in a
Fourier series ........................................ 119
5.7 Exercises ............................................. 126
References ................................................. 127
6 The fast Fourier transform (FFT) ........................... 129
6.1 Introduction .......................................... 129
6.2 Some preliminaries leading to the fast Fourier
transforms ............................................ 130
6.3 The discrete Fourier transform ........................ 143
6.4 The fast Fourier transform ............................ 149
6.4.1 An observation of the discrete Fourier
transforms ............................................ 150
6.5 Mathematical aspects of FFT ........................... 150
6.6 Reviews of some works on FFT algorithms ............... 152
6.7 Cooley-Tukey algorithms ............................... 152
6.8 Application of FFT to wave energy spectral density .... 153
6.9 Exercises ............................................. 155
References ................................................. 156
Appendix A: Table of Fourier transforms ....................... 159
A.l Fourier transforms g(y)=F{ƒ(x)}=∫∞-∞ƒ(x)e-2πixy dx ..... 159
Appendix B: Properties of impulse function (δ(x)) at a
glance ..................................................... 161
B.1 Introduction .......................................... 161
B.2 Impulse function definition ........................... 161
В.3 Properties of impulse function ........................ 161
B.3.1 Sifting property ............................... 162
B.3.2 Scaling property ............................... 162
B.3.3 Product of a δ-function by an ordinary
function ....................................... 162
B.4 Convolution property .................................. 163
В.5 δ Function as generalized limits ...................... 163
B.6 Time convolution ...................................... 164
В.7 Frequency convolution ................................. 164
Appendix C: Bibliography ...................................... 165
Subject index ................................................. 169
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