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Analysis Fourier Analysis: an Introduction by Stein and Shakarchi

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  1. Jan 25, 2013 #1

    micromass

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    Table of Contents:
    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]
    [*] Return to pointwise convergence
    [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
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