傅立叶分析导论

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 snmmation

5.2 fejer's theorem

5.3 abel means and s-ruination

5.4 the poisson 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-square 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 equidistribution 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 r

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 3

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 ra

1 preliminaries

1.1 symmetries

1.2 integration on ra

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 r3× r

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

2 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 rd

2.2 repeated integrals

2.3 the change of variables formula

2.4 spherical coordinates

3 improper integrals. integration over rd

3.1 integration of functions of moderate decrease

3.2 repeated integrals

3.3 spherical coordinates

notes and references

bibliography

symbol glossary

《傅里叶分析导论》由在国际上享有盛誉普林斯大林顿大学教授Stein撰写而成，是一部傅立叶分析的入门教材，理论与实践并重，为了便于非数专业的学生学习，全书内容简明、易懂.全书分为三部分，第一部分介绍傅立叶级数的基本理论及其在等周不等式和等分布中的应用；第二部分研究傅立叶变换及其在经典偏微分方程及Radom变换中的应用；第三部分研究有限阿贝尔群上的傅立叶分析。书中各章均有练习题及思考题。目次：傅立叶积分的起源；傅立叶级数和基本性质；傅立叶级数的收敛性；傅立叶积分的应用；IR上的傅立叶变换；IRd上的傅立叶变换；有限傅里叶分析；Dirichlet定理。