《傅裡葉分析導論》由在**上享有盛譽普林斯大林頓大學教授斯坦恩撰寫而成，是一部傅立葉分析的入門教材，理論與實踐並重，為了便於非數專業的學生學習，全書內容簡明、易懂.全書分為三部分，**部分介紹傅立葉級數的基本理論及其在等周不等式和等分布中的應用；第二部分研究傅立葉變換及其在經典偏微分方程及Radom變換中的應用；第三部分研究有限阿貝爾群上的傅立葉分析。書中各章均有練習題及思考題。目次：傅立葉積分的起源；傅立葉級數和基本性質；傅立葉級數的收斂性；傅立葉積分的應用；IR上的傅立葉變換；IRd上的傅立葉變換；有限傅裡葉分析；Dirichlet定理。

Foreword
Preface
Chapter 1. The Genesis of Fourier Analysis
1 The vibrating string
1.1 Derivation of the wave equation
1.2 Solution to the wave equation
1.3 Example: the plucked string
2 The heat equation
2.1 Derivation of the heat equation
2.2 Steady-state heat equation in the disc
3 Exercises
4 Problem
Chapter 2. Basic Properties of Fourier Series
1 Examples and formulation of the problem
1.1 Main definitions and some examples
2 Uniqueness of Fourier series
3 Convolutions
4 Good kernels
5 Cesaro and Abel summability: applications to Fourier series
5.1 Cesaro means and summation
5.2 Fejer's theorem
5.3 Abel means and summation
5.4 The Poiseon kernel and Dirichlet's problem in the
unit disc
6 Exercises
7 Problems
Chapter 3. Convergence of Fourier Series
1 Mean-square convergence of Fourier series
1.1 Vector spaces and inner products
1.2 Proof of mean-requare convergence
2 Return to pointwise convergence
2.1 A local result
2.2 A continuous function with diverging Fourier series
3 Exercises
4 Problems
Chapter 4. Some Applications of Fourier Series
1 The isoperimetric inequality
2 Weyl's eqnidistribution theorem
3 A continuous but nowhere differentiable function
4 The heat equation on the circle
5 Exercises
6 Problems
Chapter 5. The Fourier Transform on
1 Elementary theory of the Fourier transform
1.1 Integration of functions on the real line
1.2 Definition of the Fourier transform
1.3 The Schwartz space
1.4 The Fourier transform on S
1.5 The Fourier inversion
1.6 The Plancherel formula
1.7 Extension to functions of moderate decrease
1.8 The Weierstrass approximation theorem
2 Applications to some partial differential equations
2.1 The time-dependent heat equation on the real line
2.2 The steady-state heat equation in the upper half-
plane
3 The Poisson summation formula
3.1 Theta and zeta functions
3.2 Heat kernels
3.3 Poisson kernels
4 The Heisenberg uncertainty principle
5 Exercises
6 Problems
Chapter 6. The Fourier Transform on Rd
1 Preliminaries
1.1 Symmetries
1.2 Integration on ~d
2 Elementary theory of the Fourier transform
3 The wave equation in Rd ~ R
3.1 Solution in terms of Fourier transforms
3.2 The wave equation in ~3 x
3.3 The wave equation in R2 ~ R: descent
4 Radial symmetry and Bessel functions
5 The Radon transform and some of its applications
5.1 The X-ray transform in R2
5.2 The Radon transform in R3
5.3 A note about plane waves
6 Exercises
7 Problems
Chapter 7. Finite Fourier Analysis
1 Fourier analysis on Z(N)
1.1 The group Z(N)
1.2 Fourier inversion theorem and Plancherel identity on Z(N)
1.3 The fast Fourier transform
Fourier analysis on finite abelian groups
2.1 Abelian groups
2.2 Characters
2.3 The orthogonality relations
2.4 Characters as a total family
2.5 Fourier inversion and Plancherel formula
3 Exercises
4 Problems
Chapter 8. Dirichlet's Theorem
1 A little elementary number theory
1.1 The fundamental theorem of arithmetic
1.2 The infinitude of primes
2 Dirichlet's theorem
2.1 Fourier analysis, Dirichlet characters, and reduc-tion of the theorem
2.2 Dirichlet L-functions
3 Proof of the theorem
3.1 Logarithms
3.2 L-functions
3.3 Non-vanishing of the L-function
4 Exercises
5 Problems
Appendix: Integration
1 Definition of the Riemann integral
1．1 Basic properties
1. 2 Sets of measure zero and discontinuities of inte-grable functions
2 Multiple integrals
2．1 The Riemann integral in Ra
2．2 Repeatedintegrals
2．3 The change of variables formula
2．4 Spherical coordinates
3 Improper integrals．Integration over Ra
3．1 Integration of functions of moderate decrease
3．2 Repeated integrals
3．3 Spherical cootdinares
Nores and References
Bibliography
Symbol Glossary