Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

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.