Fourier analysis
Subtitle: an introduction
By: Elias M. Stein & Rami Shakarchi
Contributors: Shakarchi, Rami, author
Edition: Ying yin ban
Series: Princeton shu xue jing dian jiao cai (ying yin ban)
Publisher: Shi jie tu shu chu ban gong si
Published: 2013
Place of publication: Beijing
Language: eng
Number of pages: 311
ISBN-13: 9787510040559
Subjects: Fourier analysis
Other titles: Fuliye fen xi dao lun; Princeton lectures in analysis.
Description: This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.
Notes: Includes bibliographical references (pages 301-303) and index.

Foreword in chinese and english.
Table of contents: title: The Genesis of Fourier Analysis; title: The vibrating string; title: Derivation of the wave equation; title: Solution to the wave equation; title: Example: the plucked string; title: The heat equation; title: Derivation of the heat equation; title: Steady-state heat equation in the disc; title: Exercises; title: Problem; title: Basic Properties of Fourier Series; title: Examples and formulation of the problem; title: Main definitions and some examples; title: Uniqueness of Fourier series; title: Convulusions; title: Good kernels; title: Cesaro and Abel summability: applications to Fourier series; title: Cesaro means and summation; title: Fejer's theorem; title: Abel means and summation; title: The Poisson kernel and Dirichlet's problem in the unit disc; title: Exercises; title: Problems; title: Convergence of Fourier Series; title: Mean-square convergence of Fourier series; title: Vector spaces and inner products; title: Proof of mean-square convergence; title: Return to pointwise convergence; title: A local result; title: A continuous function with diverging Fourier series; title: Exercises; title: Problems; title: Some Applications of Fourier Series; title: The isoperimetric inequality; title: Weyl's equidistribution theorem; title: A continuous but nowhere differentiable function; title: The heat equation on the circle; title: Exercises; title: Problems; title: The Fourier Transform on R; title: Elementary theory of the Fourier transform; title: Integration of functions on the real line; title: Definition of the Fourier transform; title: The Schwartz space; title: The Fourier transform on S; title: The Fourier inversion; title: The Plancherel formula; title: Extension to functions of moderate decrease; title: The Weierstrass approximation theorem; title: Applications to some partial differential equations; title: The time-dependent heat equation on the real line; title: The steady-state heat equation in the upper half-plane; title: The Poisson summation formula; title: Theta and zeta functions; title: Heat kernels; title: Poisson kernels; title: The Heisenberg uncertainty principle; title: Exercises; title: Problems; title: The Fourier Transform on Rd; title: Preliminaries; title: Symmetries; title: Integration on Rd; title: Elementary theory of the Fourier transform; title: The wave equation in Rd x R; title: Solution in terms of Fourier transforms; title: The wave equation in R3 x R; title: The wave equation in IR2 x R: descent; title: Radial symmetry and Bessel functions; title: The Radon transform and some of its applications; title: The X-ray transform in R2; title: The Radon transform in R3; title: A note about plane waves; title: Exercises; title: Problems; title: Finite Fourier Analysis; title: Fourier analysis on Z(N); title: The group Z(N); title: Fourier inversion theorem and Plancherel identity on Z(N); title: The fast Fourier transform; title: Fourier analysis on finite abelian groups; title: Abelian groups; title: Characters; title: The orthogonality relations; title: Characters as a total family; title: Fourier inversion and Plancherel formula; title: Exercises; title: Problems; title: Dirichlet's Theorem; title: A little elementary number theory; title: The fundamental theorem of arithmetic; title: The infinitude of primes; title: Dirichlet's theorem; title: Fourier analysis, Dirichlet characters, and reduction of the theorem; title: Dirichlet L-functions; title: Proof of the theorem; title: Logarithms; title: L-functions; title: Non-vanishing of the L-function; title: Exercises; title: Problems; title: Appendix: Integration; title: Definition of the Riemann integral; title: Basic properties; title: Sets of measure zero and discontinuities of integrable functions; title: Multiple integrals; title: The Riemann integral in Rd; title: Repeated integrals; title: The change of variables formula; title: Spherical coordinates; title: Improper integrals. Integration over Rd; title: Integration of functions of moderate decrease; title: Repeated integrals; title: Spherical coordinates; title: Notes and References; title: Bibliography; title: Symbol Glossary.