Harmonic Analysis and Applications

Preface xi Prologue I---Course I xv Prologue II---Fourier transforms, Fourier series, and Discrete Fourier Transforms xix Fourier Transforms 1(70) Definitions and formal calculations 1(8) Algebraic properties of Fourier transforms 9(3) Examples 12(5) Analytic properties of Fourier transforms 17(5) Convolution 22(2) Approximate identities and examples 24(9) Pointwise inversion of the Fourier transform 33(8) Partial differential equations 41(5) Gibbs phenomenon 46(4) The L2 (R) theory 50(21) Exercises 60(11) Measures and Distribution Theory 71(82) Approximate identities and δ 71(3) Definition of distributions 74(5) Differentiation of distributions 79(9) The Fourier transform of distributions 88(8) Convolution of distributions 96(8) Operational calculus 104(6) Measure theory 110(7) Definitions from probability theory 117(8) Wiener's Generalized Harmonic Analysis (GHA) 125(8) exp {it2} 133(20) Exercises 141(12) Fourier Series 153(124) Fourier series---definitions and convergence 153(13) History of Fourier series 166(11) Integration and differentiation of Fourier series 177(8) The L1(T) and L2(T) theories 185(11) A(T) and the Wiener Inversion Theorem 196(8) Maximum entropy and spectral estimation 204(8) Prediction and spectral estimation 212(12) Discrete Fourier Transform 224(15) Fast Fourier Transform 239(9) Periodization and sampling 248(29) Exercises 259(18) A Real Analysis 277(10) B Functional Analysis 287(9) C Fourier Analysis Formulas 296(2) D Contributors to Fourier Analysis 298(7) Bibliography 305(18) Index of Notation 323(4) Index of Names 327(4) Index 331