# Analysis Fourier Analysis: an Introduction by Stein and Shakarchi

## For those who have used this book

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4. ### Strongly don't Recommend

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1. Jan 25, 2013

### micromass

Code (Text):

[LIST]
[*] Foreword
[*] Preface
[*] The Genesis of Fourier Analysis
[LIST]
[*] The vibrating string
[LIST]
[*] Derivation of the wave equation
[*] Solution to the wave equation
[*] Example: the plucked string
[/LIST]
[*] The heat equation
[LIST]
[*] Derivation of the heat equation
[*] Steady-state heat equation in the disc
[/LIST]
[*] Exercises
[*] Problem
[/LIST]
[*] Basic Properties of Fourier Series
[LIST]
[*] Examples and formulation of the problem
[LIST]
[*] Main definitions and some examples
[/LIST]
[*] Uniqueness of Fourier series
[*] Convolutions
[*] Good kernels
[*] Cesaro and Abel summability: applications to Fourier series
[LIST]
[*] Cesaro means and summation
[*] Fejer's theorem
[*] Abel means and summation
[*] The Poisson kernel and Dirichlet's problem in the
unit disc
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Convergence of Fourier Series
[LIST]
[*] Mean-square convergence of Fourier series
[LIST]
[*] Vector spaces and inner products
[*] Proof of mean-square convergence
[/LIST]
[LIST]
[*] A local result
[*] A continuous function with diverging Fourier series
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Some Applications of Fourier Series
[LIST]
[*] The isoperimetric inequality
[*] Weyl's equidistribution theorem
[*] A continuous but nowhere differentiable function
[*] The heat equation on the circle
[*] Exercises
[*] Problems
[/LIST]
[*] The Fourier Transform on R
[LIST]
[*] Elementary theory of the Fourier transform
[LIST]
[*] 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
[/LIST]
[*] Applications to some partial differential equations
[LIST]
[*] The time-dependent heat equation on the real line
[*] The steady-state heat equation in the upper half-plane
[/LIST]
[*] The Poisson summation formula
[LIST]
[*] Theta and zeta functions
[*] Heat kernels
[*] Poisson kernels
[/LIST]
[*] The Heisenberg uncertainty principle
[*] Exercises
[*] Problems
[/LIST]
[*] The Fourier Transform on R^d
[LIST]
[*] Preliminaries
[LIST]
[*] Symmetries
[*] Integration on R^d
[/LIST]
[*] Elementary theory of the Fourier transform
[*] The wave equation in R^d x R
[LIST]
[*] Solution in terms of Fourier transforms
[*] The wave equation in R^3 x R
[*] The wave equation in R^2 x R: descent
[/LIST]
[*] Radial symmetry and Bessel functions
[*] The Radon transform and some of its applications
[LIST]
[*] The X-ray transform in R^2
[*] The Radon transform in R^3
[*] A note about plane waves
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Finite Fourier Analysis
[LIST]
[*] Fourier analysis on Z(N)
[LIST]
[*] The group Z(N)
[*] Fourier inversion theorem and Plancherel identity on Z(N)
[*] The fast Fourier transform
[/LIST]
[*] Fourier analysis on finite abelian groups
[LIST]
[*] Abelian groups
[*] Characters
[*] The orthogonality relations
[*] Characters as a total family
[*] Fourier inversion and Plancherel formula
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Dirichlet's Theorem
[LIST]
[*] A little elementary number theory
[LIST]
[*] The fundamental theorem of arithmetic
[*] The infinitude of primes
[/LIST]
[*] Dirichlet's theorem
[LIST]
[*] Fourier analysis, Dirichlet characters, and reduction of the theorem
[*] Dirichlet L-functions
[/LIST]
[*] Proof of the theorem
[LIST]
[*] Logarithms
[*] L-functions
[*] Non-vanishing of the L-function
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Appendix: Integration
[LIST]
[*] Definition of the Riemann integral
[LIST]
[*] Basic properties
[*] Sets of measure zero and discontinuities of integrable functions
[/LIST]
[*] Multiple integrals
[LIST]
[*]  The Riemann integral in R^d
[*] Repeated integrals
[*] The change of variables formula
[*] Spherical coordinates
[/LIST]
[*] Improper integrals. Integration over R^d
[LIST]
[*] Integration of functions of moderate decrease
[*] Repeated integrals
[*] Spherical coordinates
[/LIST]
[/LIST]
[*] Notes and References
[*] Bibliography
[*] Symbol Glossary
[/LIST]

Last edited by a moderator: May 6, 2017