Fourier analysis : an introduction / Elias M. Stein & Rami Shakarchi.

Format:

Book

Author/Creator:

Stein, Elias M., 1931-2018.

Contributor:

Shakarchi, Rami.

Series:

Princeton lectures in analysis ; 1.

Princeton lectures in analysis ; 1

Language:

English

Subjects (All):

Fourier analysis.

Physical Description:

xvi, 311 pages : illustrations ; 24 cm.

Place of Publication:

Princeton : Princeton University Press, [2003]

Summary:

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.

Contents:

The Genesis of Fourier Analysis

The vibrating string

Derivation of the wave equation

Solution to the wave equation

Example: the plucked string

The heat equation

Derivation of the heat equation

Steady-state heat equation in the disc

Exercises

Problem

Basic Properties of Fourier Series

Examples and formulation of the problem

Main definitions and some examples

Uniqueness of Fourier series

Convulusions

Good kernels

Cesaro and Abel summability: applications to Fourier series

Cesaro means and summation

Fejer's theorem

Abel means and summation

The Poisson kernel and Dirichlet's problem in the unit disc

Problems

Convergence of Fourier Series

Mean-square convergence of Fourier series

Vector spaces and inner products

Proof of mean-square convergence

Return to pointwise convergence

A local result

A continuous function with diverging Fourier series

Some Applications of Fourier Series

The isoperimetric inequality

Weyl's equidistribution theorem

A continuous but nowhere differentiable function

The heat equation on the circle

The Fourier Transform on R

Elementary theory of the Fourier transform

Integration of functions on the real line

Definition of the Fourier transform

The Schwartz space

The Fourier transform on S

The Fourier inversion

The Plancherel formula

Extension to functions of moderate decrease

The Weierstrass approximation theorem

Applications to some partial differential equations

The time-dependent heat equation on the real line

The steady-state heat equation in the upper half-plane

The Poisson summation formula

Theta and zeta functions

Heat kernels

Poisson kernels

The Heisenberg uncertainty principle

The Fourier Transform on Rd

Preliminaries

Symmetries

Integration on Rd

The wave equation in Rd x R

Solution in terms of Fourier transforms

The wave equation in R3 x R

The wave equation in IR2 x R: descent

Radial symmetry and Bessel functions

The Radon transform and some of its applications

The X-ray transform in R2

The Radon transform in R3

A note about plane waves

Finite Fourier Analysis

Fourier analysis on Z(N)

The group Z(N)

Fourier inversion theorem and Plancherel identity on Z(N)

The fast Fourier transform

Fourier analysis on finite abelian groups

Abelian groups

Characters

The orthogonality relations

Characters as a total family

Fourier inversion and Plancherel formula

Dirichlet's Theorem

A little elementary number theory

The fundamental theorem of arithmetic

The infinitude of primes

Dirichlet's theorem

Fourier analysis, Dirichlet characters, and reduction of the theorem

Dirichlet L-functions

Proof of the theorem

Logarithms

L-functions

Non-vanishing of the L-function

Appendix: Integration

Definition of the Riemann integral

Basic properties

Sets of measure zero and discontinuities of integrable functions

Multiple integrals

The Riemann integral in Rd

Repeated integrals

The change of variables formula

Spherical coordinates

Improper integrals. Integration over Rd

Integration of functions of moderate decrease

Notes and References

Bibliography

Symbol Glossary.

Notes:

Current Copyright Fee: GBP20.00 0.

Includes bibliographical references (pages 301-303) and index.

ISBN:

069111384X

9780691113845

OCLC:

51569246

Online:

Contributor biographical information

Publisher description