Dym H., McKean H.P. Fourier Series and Integrals

London: Academic Press, 1985. — 303 p.The ideas of Fourier have made their way into every branch of mathematics and mathematical physics, from the theory of numbers to quantum mechanics. Fourier Series and Integrals focuses on the extraordinary power and flexibility of Fourier's basic series and integrals and on the astonishing variety of applications in which it is the chief tool. It presents a mathematical account of Fourier ideas on the circle and the line, on finite commutative groups, and on a few important noncommutative groups. A wide variety of exercises are placed in nearly every section as an integral part of the text.Historical Introduction
Fourier Series
The Lebesgue Integral
The Geometry of L²(Q)
The Geometry of L²(Q) Continued
Square Summable Functions on the Circle and Their Fourier Series
Summable Functions and Their Fourier Series
Gibbs' Phenomenon
Miscellaneous Applications
Applications to the Partial Differential Equations of One-Dimensional Mathematical Physics
More General Eigenfunction Expansions
Several-Dimensional Fourier Series
Fourier Integrals
Fourier Integrals
Fourier Integrals for C^?_?(R1)
Fourier Integrals for L²(R1): First Method
Fourier Integrals for L²(R1}: Second Method
Fourier Integrals for L²(R1): Third Method
Fourier Integrals for L²(R1)
Miscellaneous Applications
Heisenberg's Inequality
Band- and Time-Limited Functions
Several-Dimensional Fourier Integrals
Miscellaneous Applications of Several-Dimensional Fourier Integrals
Fourier Integrals and Complex Function Theory
A Short Course in Function Theory
Hardy's Theorem
The Paley-Wiener Theorem
Hardy Functions
Hardy Functions and Filters
Wiener-Hopf Factorization: Milne's Equation
Spitter's Identity
Hardy Functions in the Disk and Szegö's Theorem
Polynomial Approximation: The Szász-Müntz Theorem
The Prime Number Theorem
Fourier Series and Integrals on Groups
Groups
Fourier Series on the Circle
Fourier Integrals on the Line
Finite Commutative Groups
Fourier Series on a Finite Commutative Group
Gauss' Law of Quadratic Reciprocity
Noncommutative Groups
The Rotation Group
Three Convolution Algebras
Homomorphisrns of L¹(K/G/K)
Spherical Functions Are Eigenfunctions of the Laplacian
Spherical Functions Are Legendre Polynomials
Spherical Harmonics
Representations of SO(3)
The Euclidean Motion Group
SL(2,R) and the Hyperbolic Plane
Additional Reading