Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

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.

• 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 Neumann's Spectral Theorem.- Spectrum of Self-Adjoint Operators.- Quadratic Forms: Freidrich's 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.

“This book is an introduction to the modern theory of harmonic analysis and differential equations. … the book would be useful for graduate students as well as for researchers who look for applications in mathematical physics and engineering sciences.” (Sergeĭ Yu Tikhonov, Mathematical Reviews, February, 2019)