Fourier series, Fourier transform and their applications to mathematical physics
 Responsibility
 Valery Serov.
 Publication
 Cham, Switzerland : Springer, [2017]
 Physical description
 xi, 534 pages : illustrations ; 24 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 197.
Online
Available online
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA404 .S47 2017  Unknown 
More options
Description
Creators/Contributors
 Author/Creator
 Serov, Valery, author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 529530) and index.
 Contents

 Fourier series and the discrete fourier transform
 Introduction
 Formulation of fourier series
 Fourier coefficients and their properties
 Convolution and parseval's equality
 Fejér means of fourier series : uniqueness of the fourier series
 The riemannlebesgue lemma
 The fourier series of a squareintegrable function : the rieszfischer theorem
 Besov and holder spaces
 Absolute convergence : bernstein and peetre theorems
 Dirichlet kernel : pointwise and uniform convergence
 Formulation of the discrete fourier transform and its properties
 Connection between the discrete fourier transform and the fourier transform
 Some applications of the discrete fourier transform
 Applications to solving some mod el equations
 The onedimensional heat equation
 The onedimensional wave equation
 The laplace equation in a rectangle and in a disk
 Fourier transform and distributions
 Introduction
 The fourier transform in schwartz space
 The fourier transform in LP(RN), 1 < ̲p < ̲2
 Tempered distributions
 Convolutions in S and S¹
 Sobolev spaces
 Sobolev spaces on bounded domains
 Homogeneous distributions
 Fundamental solution of the helmholtz operator
 Estimates for the laplacian and hamiltonian
 Operator theory and integral equations
 Introduction
 Inner product spaces and hilbert spaces
 Symmetric operators in hilbert spaces
 John von neumann's spectral theorem
 Spectra of selfadjoint operators
 Quadratic forms : friedrichs extension
 Elliptic differential operators
 Spectral functions
 The schrodinger operator
 The magnetic schrodinger operator
 Integral operators with weak singularities : integral equations of the first and second kinds
 Volterra and singular integral equations
 Approximate methods
 Partial differential equations
 Introduction
 Local existence theory
 The laplace operator
 The dirichlet and neumann problems
 Layer potentials
 Elliptic boundary value problems
 The direct scattering problem for the helmholtz equation
 Some inverse scattering problems for the schrodinger operator
 The heat operator
 The wave operator
 References
 Index.
 Publisher's Summary
 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 halfsemester courses given at University of Oulu, this book consists of four selfcontained parts. The first part, Fourier Series and the Discrete Fourier Transform, is devoted to the classical onedimensional 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 Schroedinger and magnetic Schroedinger operations. The third part, Operator Theory and Integral Equations, is devoted mostly to the selfadjoint 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.
(source: Nielsen Book Data)9783319652610 20180403
Subjects
Bibliographic information
 Publication date
 2017
 Series
 Applied mathematical sciences, 00665452 ; volume 197
 Note
 Textbook for graduates.
 ISBN
 9783319652610 (hd.bd.)
 9783319652627 (online)
 3319652613 (hardcover)