Fourier series, Fourier transform and their applications to mathematical physics / Valery Serov.

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 inLp(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.