Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

Description

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. Front Matter ....Pages i-xi Front Matter ....Pages 1-1 Introduction (Valery Serov)....Pages 3-10 Formulation of Fourier Series (Valery Serov)....Pages 11-15 Fourier Coefficients and Their Properties (Valery Serov)....Pages 17-22 Convolution and Parseval’s Equality (Valery Serov)....Pages 23-25 Fejér Means of Fourier Series. Uniqueness of the Fourier Series. (Valery Serov)....Pages 27-31 The Riemann–Lebesgue Lemma (Valery Serov)....Pages 33-35 The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem. (Valery Serov)....Pages 37-44 Besov and Hölder Spaces (Valery Serov)....Pages 45-51 Absolute Convergence. Bernstein and Peetre Theorems. (Valery Serov)....Pages 53-58 Dirichlet Kernel. Pointwise and Uniform Convergence. (Valery Serov)....Pages 59-75 Formulation of the Discrete Fourier Transform and Its Properties. (Valery Serov)....Pages 77-84 Connection Between the Discrete Fourier Transform and the Fourier Transform. (Valery Serov)....Pages 85-92 Some Applications of the Discrete Fourier Transform. (Valery Serov)....Pages 93-98 Applications to Solving Some Model Equations (Valery Serov)....Pages 99-128 Front Matter ....Pages 129-129 Introduction (Valery Serov)....Pages 131-132 The Fourier Transform in Schwartz Space (Valery Serov)....Pages 133-141 The Fourier Transform in \(L^p(\mathbb {R}^n)\), \(1\le p\le 2\) (Valery Serov)....Pages 143-152 Tempered Distributions (Valery Serov)....Pages 153-165 Convolutions in S and \(S'\) (Valery Serov)....Pages 167-173 Sobolev Spaces (Valery Serov)....Pages 175-192 Homogeneous Distributions (Valery Serov)....Pages 193-205 Fundamental Solution of the Helmholtz Operator (Valery Serov)....Pages 207-216 Estimates for the Laplacian and Hamiltonian (Valery Serov)....Pages 217-244 Front Matter ....Pages 245-245 Introduction (Valery Serov)....Pages 247-248 Inner Product Spaces and Hilbert Spaces (Valery Serov)....Pages 249-259 Symmetric Operators in Hilbert Spaces (Valery Serov)....Pages 261-278 John von Neumann’s Spectral Theorem (Valery Serov)....Pages 279-293 Spectra of Self-Adjoint Operators (Valery Serov)....Pages 295-311 Quadratic Forms. Friedrichs Extension. (Valery Serov)....Pages 313-317 Elliptic Differential Operators (Valery Serov)....Pages 319-330 Spectral Functions (Valery Serov)....Pages 331-334 The Schrödinger Operator (Valery Serov)....Pages 335-347 The Magnetic Schrödinger Operator (Valery Serov)....Pages 349-358 Integral Operators with Weak Singularities. Integral Equations of the First and Second Kinds. (Valery Serov)....Pages 359-370 Volterra and Singular Integral Equations (Valery Serov)....Pages 371-378 Approximate Methods (Valery Serov)....Pages 379-389 Front Matter ....Pages 391-391 Introduction (Valery Serov)....Pages 393-403 Local Existence Theory (Valery Serov)....Pages 405-419 The Laplace Operator (Valery Serov)....Pages 421-435 The Dirichlet and Neumann Problems (Valery Serov)....Pages 437-449 Layer Potentials (Valery Serov)....Pages 451-469 Elliptic Boundary Value Problems (Valery Serov)....Pages 471-483 The Direct Scattering Problem for the Helmholtz Equation (Valery Serov)....Pages 485-492 Some Inverse Scattering Problems for the Schrödinger Operator (Valery Serov)....Pages 493-506 The Heat Operator (Valery Serov)....Pages 507-516 The Wave Operator (Valery Serov)....Pages 517-528 Back Matter ....Pages 529-534