Fourier Analysis of Economic Phenomena

Preface......Page 6
Contents......Page 9
 1.1 Hilbert Spaces......Page 12
 1.2 Orthonormal Systems......Page 16
 1.3 Fourier Series......Page 22
 1.4 Completeness of Orthonormal Systems......Page 25
 References......Page 32
2 Convergence of Classical Fourier Series......Page 33
 2.1 Dirichlet Integrals......Page 34
 2.2 Dini, Jordan Tests......Page 38
 2.3 Almost Everywhere Convergence: Historical Survey......Page 44
 2.4 Uniform Convergence......Page 47
 2.5 Fejér Integral and (C, 1)-summability......Page 50
 References......Page 54
 3.1 Fourier Integrals......Page 56
 3.2 Fourier Transforms on L1( R, C)......Page 61
 3.3 Application: Heat Equation......Page 69
 References......Page 71
 4.1 Fourier Transforms of Rapidly Decreasing Functions......Page 73
 4.2 Fourier Transforms on L2(R, C)......Page 80
 4.3 Application: Integral Equations of Convolution Type......Page 83
 4.4 Fourier Transforms of Tempered Distributions......Page 86
 4.5 Fourier Transforms on L2(R, C) Revisited......Page 95
 4.6 Periodic Distributions......Page 100
 References......Page 107
 5.1 Shift Operators......Page 109
 5.2 Summability Kernels on [-π, π]......Page 113
 5.3 Spectral Synthesis on [-π, π]......Page 117
 5.4 Summability Kernels on R......Page 119
 5.5 Spectral Synthesis on R: Inverse Fourier Transforms on L1......Page 123
 References......Page 127
 6.1 Radon Measures......Page 128
 6.2 Fourier Coefficients of Measures (1)......Page 129
 6.3 Fourier Coefficients of Measures (2)......Page 133
 6.4 Herglotz\'s Theorem......Page 136
 6.5 Fourier Transforms of Measures......Page 143
 6.6 Bochner\'s Theorem......Page 153
 6.7 Convolutions of Measures......Page 161
 6.8 Wiener\'s Theorem......Page 165
 References......Page 170
 7.1 Lax–Milgram Theorem......Page 171
 7.2 Conjugate Operators and Projections......Page 175
 7.3 Unitary Operators......Page 181
 7.4 Resolution of the Identity......Page 183
 7.5 Spectral Representation of Unitary Operators......Page 186
 7.6 Stone\'s Theorem......Page 193
 References......Page 197
8 Periodic Weakly Stationary Processes......Page 198
 8.1 Stochastic Processes of Second Order......Page 199
 8.2 Weakly Stationary Stochastic Processes......Page 205
 8.3 Periodicity of Weakly Stationary Stochastic Process......Page 215
 8.4 Orthogonal Measures......Page 221
 8.5 Spectral Representation of Weakly Stationary Processes......Page 226
 8.6 Spectral Density Functions......Page 232
 8.7 A Note on Slutsky\'s Work......Page 243
 References......Page 248
 9.1 Almost Periodic Functions......Page 250
 9.2 AP(R,C) as a Closed Subalgebra of L∞(R,C)......Page 252
 9.3 Spectrum of Almost Periodic Functions......Page 261
 9.4 Fourier Series of Almost Periodic Functions......Page 271
 9.5 Almost Periodic Weakly Stationary Stochastic Processes......Page 279
 References......Page 282
 10.1 Direct Sums and Projections......Page 284
 10.2 Fredholm Operators: Definitions and Examples......Page 295
 10.3 Parametrix......Page 298
 10.4 Product of Fredholm Operators......Page 300
 10.5 Stability of Indices......Page 303
 References......Page 304
11 Hopf Bifurcation Theorem......Page 305
 11.1 Ljapunov–Schmidt Reduction Method......Page 307
 11.2 Abstract Hopf Bifurcation Theorem......Page 309
 11.3 Classical Hopf Bifurcation for Ordinary Differential Equations......Page 313
 11.4 Smoothness of F......Page 316
 11.5 dim V=2......Page 319
 11.6 codim R=2......Page 321
 11.7 Linear Independence of PMv* and PNv* (1)......Page 325
 11.8 Linear Independence of PMv* and PNv* (2)......Page 326
 11.9 Hopf Bifurcation in Cr......Page 332
 11.10 Kaldorian Business Fluctuations......Page 334
 11.11 Ljapunov\'s Center Theorem......Page 339
 References......Page 348
 A.1 Complex Exponential Function......Page 350
 A.2 Imaginary Exponential Function......Page 352
 A.3 Torus R/2πZ......Page 355
 A.4 A Homomorphism of R into U......Page 357
 A.5 Functions and σ-Fields on the Torus......Page 358
 References......Page 359
 B.1 Inductive Limit Topology......Page 360
 B.2 Duals of Locally Convex Spaces......Page 369
 References......Page 380
 C.1 The Space D......Page 382
 C.2 Examples of Test Functions and an Approximation Theorem......Page 387
 C.3 Distributions: Definition and Examples......Page 391
 C.4 Differentiation of Distributions......Page 395
 C.5 Topologies on the Space D(Ω)\' of Distributions......Page 399
 References......Page 405
Addendum......Page 406
 References......Page 407
Name Index......Page 408
Subject Index......Page 410