An edition of Fourier analysis
(2003)
Fourier analysis
an introduction
Ying yin ban
by Elias M. Stein
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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.
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Fourier analysisShowing 2 featured editions. View all 2 editions?
| Edition | Availability |
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Fourier analysis: an introduction
2013, Shi jie tu shu chu ban gong si
in English
- Ying yin ban
7510040558 9787510040559
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Fourier analysis: an introduction
2003, Princeton University Press
in English
069111384X 9780691113845
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Book Details
Table of Contents
The Genesis of Fourier Analysis
The vibrating string
Derivation of the wave equation
Solution to the wave equation
Example: the plucked string
The heat equation
Derivation of the heat equation
Steady-state heat equation in the disc
Exercises
Problem
Basic Properties of Fourier Series
Examples and formulation of the problem
Main definitions and some examples
Uniqueness of Fourier series
Convulusions
Good kernels
Cesaro and Abel summability: applications to Fourier series
Cesaro means and summation
Fejer's theorem
Abel means and summation
The Poisson kernel and Dirichlet's problem in the unit disc
Exercises
Problems
Convergence of Fourier Series
Mean-square convergence of Fourier series
Vector spaces and inner products
Proof of mean-square convergence
Return to pointwise convergence
A local result
A continuous function with diverging Fourier series
Exercises
Problems
Some Applications of Fourier Series
The isoperimetric inequality
Weyl's equidistribution theorem
A continuous but nowhere differentiable function
The heat equation on the circle
Exercises
Problems
The Fourier Transform on R
Elementary theory of the Fourier transform
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
Applications to some partial differential equations
The time-dependent heat equation on the real line
The steady-state heat equation in the upper half-plane
The Poisson summation formula
Theta and zeta functions
Heat kernels
Poisson kernels
The Heisenberg uncertainty principle
Exercises
Problems
The Fourier Transform on Rd
Preliminaries
Symmetries
Integration on Rd
Elementary theory of the Fourier transform
The wave equation in Rd x R
Solution in terms of Fourier transforms
The wave equation in R3 x R
The wave equation in IR2 x R: descent
Radial symmetry and Bessel functions
The Radon transform and some of its applications
The X-ray transform in R2
The Radon transform in R3
A note about plane waves
Exercises
Problems
Finite Fourier Analysis
Fourier analysis on Z(N)
The group Z(N)
Fourier inversion theorem and Plancherel identity on Z(N)
The fast Fourier transform
Fourier analysis on finite abelian groups
Abelian groups
Characters
The orthogonality relations
Characters as a total family
Fourier inversion and Plancherel formula
Exercises
Problems
Dirichlet's Theorem
A little elementary number theory
The fundamental theorem of arithmetic
The infinitude of primes
Dirichlet's theorem
Fourier analysis, Dirichlet characters, and reduction of the theorem
Dirichlet L-functions
Proof of the theorem
Logarithms
L-functions
Non-vanishing of the L-function
Exercises
Problems
Appendix: Integration
Definition of the Riemann integral
Basic properties
Sets of measure zero and discontinuities of integrable functions
Multiple integrals
The Riemann integral in Rd
Repeated integrals
The change of variables formula
Spherical coordinates
Improper integrals. Integration over Rd
Integration of functions of moderate decrease
Repeated integrals
Spherical coordinates
Notes and References
Bibliography
Symbol Glossary.
Edition Notes
Includes bibliographical references (pages 301-303) and index.
Foreword in chinese and english.
Classifications
The Physical Object
ID Numbers
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