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.- Introduction.- Formulation of Fourier Series.- Fourier Coefficients and their Properties.- Convolution and Parseval Equality.- Fejer Means of Fourier Series: Uniqueness of the Fourier Series.- Riemann-Lebesgue Lemma.- Fourier Series of Square-Integrable Function: Riesz-Fischer Theorem.- Besov and Holder Spaces.- Absolute Convergence: Bernstein and Peetre Theorems.- Dirichlet Kernel: Pointwise and Uniform Congergence.- Formulation of Discrete Fourier Transform and its Properties.- Connection Between the Discrete Fourier Transform and the Fourier Transform.- Some Applications of Discrete Fourier Transform.- Applications to Solving Some Model Equations.- Part II: Fourier Transform and Distributions.- Introduction.- Fourier Transform in Schwartz Space.- Fourier Transform in Lp(Rn)1 p 2.- Tempered Distributions.- Convolutions in S and S^1.- Sobolev Spaces.- Homogeneous Distributions.- Fundamental Solution of the Helmholtz Operator.- Estimates for Laplacian and Hamiltonian.- Part III: Operator Theory and Integral Equations.- Introduction.- Inner Product Spaces and Hilbert Spaces.- Symmetric Operators in Hilbert Spaces.- J. von Neumanns Spectral Theorem.- Spectrum of Self-Adjoint Operators.- Quadratic Forms: Freidrichs Extension.- Elliptic Differential Operators.- Spectral Function.- Schrodinger Operator.- Magnetic Schrodinger Operator.- Integral Operators with Weak Singularities: Integral Equations of the First and Second Kind.- Volterra and Singular Integral Equations.- Approximate Methods.- Part IV: Partial Differential Equations.- Introduction.- Local Existence Theory.- The Laplace Operator.- The Dirichlet and Neumman Problems.- Layer Potentials.- Elliptic Boundary Value Problems.- Direct Scattering Problem for Helmholtz Equation.- Some Inverse Scattering Problems for the Schrodinger Operator.- The Heat Operator.- The Wave Operator.