Fourier Analysis: An Introduction (Princeton Lectures in Analysis, Volume I)

✍ Scribed by Elias M. Stein, Rami Shakarchi

• Publisher: Princeton University Press
• Year: 2003
• Tongue: English
• Leaves: 326
• Series: Princeton Lectures in Analysis, 1
• Edition: 1
• Category: Library

✦ Synopsis

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.


The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.


In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest.


The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

✦ Table of Contents

Book I
Cover
Half-Title
Title
Copyright
Authors’ Dedications
Foreword
Preface to Book I
Contents
Chapter 1. The Genesis of Fourier Analysis
1.1 The vibrating string
Simple harmonic motion
Standing and traveling waves
Harmonics and superposition of tones
1.1.1 Derivation of the wave equation
1.1.2 Solution to the wave equation
Traveling waves
Superposition of standing waves
1.1.3 Example: the plucked string
1.2 The heat equation
1.2.1 Derivation of the heat equation
1.2.2 Steady-state heat equation in the disc
1.3 Exercises
1.4 Problem
Chapter 2. Basic Properties of Fourier Series
2.1 Examples and formulation of the problem
Everywhere continuous functions
Piecewise continuous functions
Riemann integrable functions
Functions on the circle
2.1.1 Main definitions and some examples
2.2 Uniqueness of Fourier series
2.3 Convolutions
2.4 Good kernels
2.5 Cesàro and Abel summability: applications to Fourier series
2.5.1 Cesàro means and summation
2.5.2 Fejér’s theorem
2.5.3 Abel means and summation
2.5.4 The Poisson kernel and Dirichlet’s problem in the unit disc
2.6 Exercises
2.7 Problems
Chapter 3. Convergence of Fourier Series
3.1 Mean-square convergence of Fourier series
3.1.1 Vector spaces and inner products
Preliminaries on vector spaces
Two important examples
3.1.2 Proof of mean-square convergence
3.2 Return to pointwise convergence
3.2.1 A local result
3.2.2 A continuous function with diverging Fourier series
3.3 Exercises
3.4 Problems
Chapter 4. Some Applications of Fourier Series
4.1 The isoperimetric inequality
Curves, length and area
Statement and proof of the isoperimetric inequality
4.2 Weyl’s equidistribution theorem
The reals modulo the integers
4.3 A continuous but nowhere differentiable function
4.4 The heat equation on the circle
4.5 Exercises
4.6 Problems
Chapter 5. The Fourier Transform on R
5.1 Elementary theory of the Fourier transform
5.1.1 Integration of functions on the real line
5.1.2 Definition of the Fourier transform
5.1.3 The Schwartz space
5.1.4 The Fourier transform on S
The Gaussians as good kernels
5.1.5 The Fourier inversion
5.1.6 The Plancherel formula
5.1.7 Extension to functions of moderate decrease
5.1.8 The Weierstrass approximation theorem
5.2 Applications to some partial differential equations
5.2.1 The time-dependent heat equation on the real line
5.2.2 The steady-state heat equation in the upper half-plane
5.3 The Poisson summation formula
5.3.1 Theta and zeta functions
5.3.2 Heat kernels
5.3.3 Poisson kernels
5.4 The Heisenberg uncertainty principle
5.5 Exercises
5.6 Problems
Chapter 6. The Fourier Transform on Rd
6.1 Preliminaries
6.1.1 Symmetries
6.1.2 Integration on Rd
Polar coordinates
6.2 Elementary theory of the Fourier transform
6.3 The wave equation in Rd × R
6.3.1 Solution in terms of Fourier transforms
6.3.2 The wave equation in R3 × R
Huygens principle
6.3.3 The wave equation in R2 × R: descent
6.4 Radial symmetry and Bessel functions
6.5 The Radon transform and some of its applications
6.5.1 The X-ray transform in R2
6.5.2 The Radon transform in R3
6.5.3 A note about plane waves
6.6 Exercises
6.7 Problems
Chapter 7. Finite Fourier Analysis
7.1 Fourier analysis on Z(N)
7.1.1 The group Z(N)
7.1.2 Fourier inversion theorem and Plancherel identity on Z(N)
7.1.3 The fast Fourier transform
7.2 Fourier analysis on finite abelian groups
7.2.1 Abelian groups
Examples of abelian groups
The group Z∗(q)
7.2.2 Characters
7.2.3 The orthogonality relations
7.2.4 Characters as a total family
7.2.5 Fourier inversion and Plancherel formula
7.3 Exercises
7.4 Problems
Chapter 8. Dirichlet’s Theorem
8.1 A little elementary number theory
8.1.1 The fundamental theorem of arithmetic
8.1.2 The infinitude of primes
The zeta function and its Euler product
8.2 Dirichlet’s theorem
8.2.1 Fourier analysis, Dirichlet characters, and reduction of the
8.2.2 Dirichlet L-functions
Historical digression
8.3 Proof of the theorem
8.3.1 Logarithms
8.3.2 L-functions
8.3.3 Non-vanishing of the L-function
Case I: complex Dirichlet characters
Case II: real Dirichlet characters
8.4 Exercises
8.5 Problems
Appendix : Integration
A.1 Definition of the Riemann integral
A.1.1 Basic properties
A.1.2 Sets of measure zero and discontinuities of integrable func-
A.2 Multiple integrals
A.2.1 The Riemann integral in Rd
Definitions
A.2.2 Repeated integrals
A.2.3 The change of variables formula
A.2.4 Spherical coordinates
A.3 Improper integrals. Integration over Rd
A.3.1 Integration of functions of moderate decrease
A.3.2 Repeated integrals
A.3.3 Spherical coordinates
Notes and References
Bibliography
Symbol Glossary
Index