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 6 Contents 9 1 Fourier Series on Hilbert Spaces 12 1.1 Hilbert Spaces 12 1.2 Orthonormal Systems 16 1.3 Fourier Series 22 1.4 Completeness of Orthonormal Systems 25 References 32 2 Convergence of Classical Fourier Series 33 2.1 Dirichlet Integrals 34 2.2 Dini, Jordan Tests 38 2.3 Almost Everywhere Convergence: Historical Survey 44 2.4 Uniform Convergence 47 2.5 Fejér Integral and (C, 1)-summability 50 References 54 3 Fourier Transforms (I) 56 3.1 Fourier Integrals 56 3.2 Fourier Transforms on L1( R, C) 61 3.3 Application: Heat Equation 69 References 71 4 Fourier Transforms (II) 73 4.1 Fourier Transforms of Rapidly Decreasing Functions 73 4.2 Fourier Transforms on L2(R, C) 80 4.3 Application: Integral Equations of Convolution Type 83 4.4 Fourier Transforms of Tempered Distributions 86 4.5 Fourier Transforms on L2(R, C) Revisited 95 4.6 Periodic Distributions 100 References 107 5 Summability Kernels and Spectral Synthesis 109 5.1 Shift Operators 109 5.2 Summability Kernels on [-π, π] 113 5.3 Spectral Synthesis on [-π, π] 117 5.4 Summability Kernels on R 119 5.5 Spectral Synthesis on R: Inverse Fourier Transforms on L1 123 References 127 6 Fourier Transforms of Measures 128 6.1 Radon Measures 128 6.2 Fourier Coefficients of Measures (1) 129 6.3 Fourier Coefficients of Measures (2) 133 6.4 Herglotz's Theorem 136 6.5 Fourier Transforms of Measures 143 6.6 Bochner's Theorem 153 6.7 Convolutions of Measures 161 6.8 Wiener's Theorem 165 References 170 7 Spectral Representation of Unitary Operators 171 7.1 Lax–Milgram Theorem 171 7.2 Conjugate Operators and Projections 175 7.3 Unitary Operators 181 7.4 Resolution of the Identity 183 7.5 Spectral Representation of Unitary Operators 186 7.6 Stone's Theorem 193 References 197 8 Periodic Weakly Stationary Processes 198 8.1 Stochastic Processes of Second Order 199 8.2 Weakly Stationary Stochastic Processes 205 8.3 Periodicity of Weakly Stationary Stochastic Process 215 8.4 Orthogonal Measures 221 8.5 Spectral Representation of Weakly Stationary Processes 226 8.6 Spectral Density Functions 232 8.7 A Note on Slutsky's Work 243 References 248 9 Almost Periodic Functions and Stochastic Processes 250 9.1 Almost Periodic Functions 250 9.2 AP(R,C) as a Closed Subalgebra of L∞(R,C) 252 9.3 Spectrum of Almost Periodic Functions 261 9.4 Fourier Series of Almost Periodic Functions 271 9.5 Almost Periodic Weakly Stationary Stochastic Processes 279 References 282 10 Fredholm Operators 284 10.1 Direct Sums and Projections 284 10.2 Fredholm Operators: Definitions and Examples 295 10.3 Parametrix 298 10.4 Product of Fredholm Operators 300 10.5 Stability of Indices 303 References 304 11 Hopf Bifurcation Theorem 305 11.1 Ljapunov–Schmidt Reduction Method 307 11.2 Abstract Hopf Bifurcation Theorem 309 11.3 Classical Hopf Bifurcation for Ordinary Differential Equations 313 11.4 Smoothness of F 316 11.5 dim V=2 319 11.6 codim R=2 321 11.7 Linear Independence of PMv* and PNv* (1) 325 11.8 Linear Independence of PMv* and PNv* (2) 326 11.9 Hopf Bifurcation in Cr 332 11.10 Kaldorian Business Fluctuations 334 11.11 Ljapunov's Center Theorem 339 References 348 Appendix A Exponential Function eiθ 350 A.1 Complex Exponential Function 350 A.2 Imaginary Exponential Function 352 A.3 Torus R/2πZ 355 A.4 A Homomorphism of R into U 357 A.5 Functions and σ-Fields on the Torus 358 References 359 Appendix B Topics from Functional Analysis 360 B.1 Inductive Limit Topology 360 B.2 Duals of Locally Convex Spaces 369 References 380 Appendix C Theory of Distributions 382 C.1 The Space D 382 C.2 Examples of Test Functions and an Approximation Theorem 387 C.3 Distributions: Definition and Examples 391 C.4 Differentiation of Distributions 395 C.5 Topologies on the Space D(Ω)' of Distributions 399 References 405 Addendum 406 References 407 Name Index 408 Subject Index 410 Front Matter ....Pages i-xi Fourier Series on Hilbert Spaces (Toru Maruyama)....Pages 1-21 Convergence of Classical Fourier Series (Toru Maruyama)....Pages 23-45 Fourier Transforms (I) (Toru Maruyama)....Pages 47-63 Fourier Transforms (II) (Toru Maruyama)....Pages 65-100 Summability Kernels and Spectral Synthesis (Toru Maruyama)....Pages 101-119 Fourier Transforms of Measures (Toru Maruyama)....Pages 121-163 Spectral Representation of Unitary Operators (Toru Maruyama)....Pages 165-191 Fourier Analysis of Periodic Weakly Stationary Processes (Toru Maruyama)....Pages 193-244 Almost Periodic Functions and Weakly Stationary Stochastic Processes (Toru Maruyama)....Pages 245-278 Fredholm Operators (Toru Maruyama)....Pages 279-299 Hopf Bifurcation Theorem (Toru Maruyama)....Pages 301-345 Back Matter ....Pages 347-410