Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

This text serves as an introduction to the modern theory of analysis and differential equations with applications in mathematical physics and engineering sciences. Having outgrown from a series of half-semester courses given at University of Oulu, this book consists of four self-contained parts. The first part, Fourier Series and the Discrete Fourier Transform, is devoted to the classical one-dimensional trigonometric Fourier series with some applications to PDEs and signal processing. The second part, Fourier Transform and Distributions, is concerned with distribution theory of L. Schwartz and its applications to the Schrödinger and magnetic Schrödinger operations. The third part, Operator Theory and Integral Equations, is devoted mostly to the self-adjoint but unbounded operators in Hilbert spaces and their applications to integral equations in such spaces. The fourth and final part, Introduction to Partial Differential Equations, serves as an introduction to modern methods for classical theory of partial differential equations. Complete with nearly 250 exercises throughout, this text is intended for graduate level students and researchers in the mathematical sciences and engineering.

Part I Fourier Series and the Discrete Fourier Transform 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Formulation of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Fourier Coefficients and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Convolution and Parseval’s Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Fej´er Means of Fourier Series. Uniqueness of the Fourier Series. . . . . 27 6 The Riemann–Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7 The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8 Besov and H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 9 Absolute convergence. Bernstein and Peetre Theorems. . . . . . . . . . . . . . 53 10 Dirichlet Kernel. Pointwise and Uniform Convergence. . . . . . . . . . . . . . 59 11 Formulation of the Discrete Fourier Transform and Its Properties. . . . 77 12 Connection Between the Discrete Fourier Transform and the Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 13 Some Applications of the Discrete Fourier Transform. . . . . . . . . . . . . . . 93 14 Applications to Solving Some Model Equations . . . . . . . . . . . . . . . . . . . . 99 14.1 The One-Dimensional Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 99 14.2 The One-Dimensional Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 113 14.3 The Laplace Equation in a Rectangle and in a Disk . . . . . . . . . . . . . 121 Part II Fourier Transform and Distributions 15 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 16 The Fourier Transform in Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . . 133 17 The Fourier Transform in Lp(Rn), 1 ≤ p ≤ 2. . . . . . . . . . . . . . . . . . . . . . . 143 18 Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 19 Convolutions in S and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 20 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 20.1 Sobolev spaces on bounded domains . . . . . . . . . . . . . . . . . . . . . . . . . 188 21 Homogeneous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 22 Fundamental Solution of the Helmholtz Operator . . . . . . . . . . . . . . . . . . 207 23 Estimates for the Laplacian and Hamiltonian . . . . . . . . . . . . . . . . . . . . . 217 Part III Operator Theory and Integral Equations 24 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 25 Inner Product Spaces and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 249 26 Symmetric Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 27 John von Neumann’s spectral theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 28 Spectra of Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 29 Quadratic Forms. Friedrichs Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . 313 30 Elliptic Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 31 Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 32 The Schr¨odinger Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 33 The Magnetic Schr¨odinger Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 34 Integral Operators withWeak Singularities. Integral Equations of the First and Second Kinds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 35 Volterra and Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . 371 36 Approximate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Part IV Partial Differential Equations 37 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 38 Local Existence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 39 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 40 The Dirichlet and Neumann Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 41 Layer Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 42 Elliptic Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 43 The Direct Scattering Problem for the Helmholtz Equation . . . . . . . . . 485 44 Some Inverse Scattering Problems for the Schr¨odinger Operator . . . . 493 45 The Heat Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 46 The Wave Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Valery Serov is Professor of Mathematics at the University of Oulu. Professor Serov received his PhD in Applied Mathematics in 1979 from Lomonosov Moscow State University. He has over 120 publications, including 3 textbooks published in Russian.

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