Fourier Analysis of Economic Phenomena (Monographs in Mathematical Economics Book 2)

"This is the first monograph that discusses in detail the interactions between Fourier analysis and dynamic economic theories, in particular, business cycles. Many economic theories have analyzed cyclical behaviors of economic variables. In this book, the focus is on a couple of trials: (1) the Kaldor theory and (2) the Slutsky effect. The Kaldor theory tries to explain business fluctuations in terms of nonlinear, 2nd-order ordinary differential equations (ODEs). In order to explain periodic behaviors of a solution, the Hopf-bifurcation theorem frequently plays a key role. Slutsky's idea is to look at the periodic movement as an overlapping effect of random shocks. The Slutsky process is a weakly stationary process, the periodic (or almost periodic) behavior of which can be analyzed by the Bochner theorem. The goal of this book is to give a comprehensive and rigorous justification of these ideas. Therefore, the aim is first to give a complete theory that supports the Hopf theorem and to prove the existence of periodic solutions of ODEs; and second to explain the mathematical structure of the Bochner theorem and its relation to periodic (or almost periodic) behaviors of weakly stationary processes. Although these two targets are the principal ones, a large number of results from Fourier analysis must be prepared in order to reach these goals. The basic concepts and results from classical as well as generalized Fourier analysis are provided in a systematic way. Prospective readers are assumed to have sufficient knowledge of real, complex analysis. However, necessary economic concepts are explained in the text, making this book accessible even to readers without a background in economics"--Page 4 of cover 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