| Preface | p. xi |
| Introduction | p. 1 |
| Lebesgue integration | p. 4 |
| Introduction | p. 4 |
| Riemann integration | p. 4 |
| Null sets | p. 5 |
| The Lebesgue integral | p. 6 |
| Nomenclature | p. 8 |
| Conditions for integrability; measurability | p. 9 |
| Functions in L[[superscript p]] | p. 12 |
| Integrals in several dimensions | p. 13 |
| Alternative approaches | p. 14 |
| Some useful theorems | p. 15 |
| The Minkowski inequality | p. 15 |
| Holder's theorem | p. 16 |
| Young's theorem | p. 17 |
| The Fubini and Tonelli theorems | p. 18 |
| Two theorems of Lebesgue | p. 19 |
| Absolute and uniform continuity | p. 20 |
| The Riemann--Lebesgue theorem | p. 23 |
| Convergence of sequences of functions | p. 24 |
| Introduction | p. 24 |
| Pointwise convergence | p. 24 |
| Bounded, dominated and monotone convergence | p. 25 |
| Uniform convergence | p. 26 |
| Convergence in the mean | p. 27 |
| Cauchy sequences | p. 29 |
| Local averages and convolution kernels | p. 30 |
| Introduction | p. 30 |
| Lebesgue points | p. 31 |
| Approximate convolution identities | p. 33 |
| The Dirichlet kernel and Dirichlet points | p. 35 |
| The functions of du Bois-Reymond and of Fejer | p. 36 |
| Carleson's theorem | p. 37 |
| Kolmogoroff's theorem | p. 37 |
| The Dirichlet conditions | p. 38 |
| Jordan's theorem | p. 39 |
| Dini's theorem | p. 41 |
| The de la Vallee-Poussin test | p. 43 |
| Some general remarks on Fourier transformation | p. 44 |
| Introduction | p. 44 |
| The definition of the Fourier transform | p. 44 |
| Sufficient conditions for transformability | p. 47 |
| Necessary conditions for transformability | p. 49 |
| Fourier theorems for good functions | p. 51 |
| Introduction | p. 51 |
| Inversion, differentiation and convolution theorems | p. 52 |
| Good functions of bounded support | p. 55 |
| Fourier theorems in L[[superscript p]] | p. 60 |
| Basic theorems and definitions | p. 60 |
| More inversion theorems in L[[superscript p]] | p. 63 |
| Convolution and product theorems in L[[superscript p]] | p. 71 |
| Uncertainty principle and bandwidth theorem | p. 75 |
| The sampling theorem | p. 77 |
| Hilbert transforms and causal functions | p. 78 |
| Fourier theorems for functions outside L[[superscript p]] | p. 81 |
| Introduction | p. 81 |
| Functions in class K | p. 82 |
| Convolutions and products in K | p. 85 |
| Functions outside K | p. 87 |
| Miscellaneous theorems | p. 91 |
| Differentiation and integration | p. 91 |
| The Gibbs phenomenon | p. 93 |
| Complex Fourier transforms | p. 95 |
| Positive-definite and distribution functions | p. 99 |
| Power spectra and Wiener's theorems | p. 102 |
| Introduction | p. 102 |
| The autocorrelation function | p. 104 |
| The spectrum and spectral density | p. 106 |
| Discrete spectra | p. 109 |
| Continuous spectra | p. 113 |
| Miscellaneous theorems | p. 114 |
| Generalized functions | p. 118 |
| Introduction | p. 118 |
| The definition of functionals in S' | p. 119 |
| Basic theorems | p. 123 |
| Examples of generalized functions | p. 127 |
| Fourier transformation of generalized functions I | p. 135 |
| Definition of the transform | p. 135 |
| Simple properties of the transform | p. 136 |
| Examples of Fourier transforms | p. 137 |
| The convolution and product of functionals | p. 139 |
| Fourier transformation of generalized functions II | p. 145 |
| Functionals of types D' and Z' | p. 145 |
| Fourier transformation of functionals in D' | p. 149 |
| Transformation of products and convolutions in D' | p. 152 |
| Fourier series | p. 155 |
| Fourier coefficients of a periodic function | p. 155 |
| The convergence of Fourier series | p. 156 |
| Summability of Fourier series | p. 158 |
| Mean convergence of Fourier series | p. 159 |
| Sampling theorems | p. 162 |
| Differentiation and integration of Fourier series | p. 164 |
| Products and convolutions | p. 166 |
| Generalized Fourier series | p. 170 |
| Introduction | p. 170 |
| Generalized Fourier coefficients | p. 171 |
| The Fourier formulae | p. 172 |
| Differentiation, repetition and sampling | p. 173 |
| Products and convolutions | p. 175 |
| Bibliography | p. 177 |
| Index | p. 181 |
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