Körner T.W. Fourier Analysis

Cambridge University Press, 2022. — xiv, 592 p. — (Cambridge Mathematical Library). — ISBN 978-1-009-23005-6, 978-1-009-23006-3.Fourier analysis is a subject that was born in physics but grew up in mathematics. Now it is part of the standard repertoire for mathematicians, physicists and engineers. In most books, this diversity of interest is often ignored, but here Dr Körner has provided a shop-window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy, and electrical engineering. Each application is placed in perspective by a short essay. The prerequisites are few (the reader with knowledge of second or third year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining. In short, this stimulating account will be welcomed by all who like to read about more than the bare bones of a subject. For them this will be a meaty guide to Fourier analysis.Foreword by Terence Tao
Preface
Fourier Series
Introduction
Proof of Fejér’s theorem
Weyl’s equidistribution theorem
The Weierstrass polynomial approximation theorem
A second proof of Weierstrass’s theorem
Hausdorff’s moment problem
The importance of linearity
Compass and tides
The simplest convergence theorem
The rate of convergence
A nowhere differentiable function
Reactions
Monte Carlo methods
Mathematical Brownian motion
Pointwise convergence
Behaviour at points of discontinuity I
Behaviour at points of discontinuity II
A Fourier series divergent at a point
Pointwise convergence, the answer
Some Differential Equations
The undisturbed damped oscillator does not explode
The disturbed damped linear oscillator does not explode
Transients
The linear damped oscillator with periodic input
A non-linear oscillator I
A non-linear oscillator II
A non-linear oscillator III
Poisson summation
Dirichlet’s problem for the disc
Potential theory with smoothness assumptions
An example of Hadamard
Potential theory without smoothness assumptions
Orthogonal Series
Mean square approximation I
Mean square approximation II
Mean square convergence
The isoperimetric problem I
The isoperimetric problem II
The Sturm–Liouville equation I
Liouville
The Sturm–Liouville equation II
Orthogonal polynomials
Gaussian quadrature
Linkages
Tchebychev and uniform approximation I
The existence of the best approximation
Tchebychev and uniform approximation II
Fourier Transforms
Introduction
Change in the order of integration I
Change in the order of integration II
Fejér’s theorem for Fourier transforms
Sums of independent random variables
Convolution
Convolution on T
Differentiation under the integral
Lord Kelvin
The heat equation
The age of the earth I
The age of the earth II
The age of the earth III
Weierstrass’s proof of Weierstrass’s theorem
The inversion formula
Simple discontinuities
Heat flow in a semi-infinite rod
A second approach
The wave equation
The transatlantic cable I
The transatlantic cable II
Uniqueness for the heat equation I
Uniqueness for the heat equation II
The law of errors
The central limit theorem I
The central limit theorem II
Further Developments
Stability and control
Instability
The Laplace transform
Deeper properties
Poles and stability
A simple time delay equation
An exception to a rule
Many dimensions
Sums of random vectors
A chi squared test
Haldane on fraud
An example of outstanding statistical treatment I
An example of outstanding statistical treatment II
An example of outstanding statistical treatment III
Will a random walk return?
Will a Brownian motion return?
Analytic maps of Brownian motion
Will a Brownian motion tangle?
La Famille Picard va á Monte Carlo
Other Directions
The future of mathematics viewed from 1800
Who was Fourier? I
Who was Fourier? II
Why do we compute?
The diameter of stars
What do we compute?
Fourier analysis on the roots of unity
How do we compute?
How fast can we multiply?
What makes a good code?
A little group theory
A good code?
A little more group theory
ourier analysis on finite Abelian groups
A formula of Euler
An idea of Dirichlet
Primes in some arithmetical progressions
Extension from real to complex variable
Primes in general arithmetical progressions
A word from our founderAppendix A: The circle T
Appendix B: Continuous function on closed bounded sets
Appendix C: Weakening hypotheses
Appendix D: Ode to a galvanometer
Appendix E: The principle of the argument
Appendix F: Chase the constant
Appendix G: Are share prices in Brownian motion?
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