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Fourier Series and Integrals

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Author(s): H Dym, HP McKean
Series: Probability and Mathematical Statistics
Edition: revised
Publisher: Academic Press
Year: 1985

Language: English
Commentary: 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.
Pages: 0
Tags: harmonic analysis, group theory, representation theory, measure theory

Preface/10,Black,notBold,notItalic,open,FitWidth,-1
Historical Introduction/12,Black,notBold,notItalic,open,FitWidth,-1
Chapter 1. Fourier Series/16,Black,notBold,notItalic,open,FitWidth,-1
1.1 The Lebesgue Integral/16,Black,notBold,notItalic,open,FitWidth,-1
1.2 The Geometry of L²(Q)/23,Black,notBold,notItalic,open,FitWidth,-1
1.3 The Geometry of L²(Q) Continued/33,Black,notBold,notItalic,open,TopLeftZoom,449,0,0.0
1.4 Square Summable Functions on the Circle and Their Fourier Series/40,Black,notBold,notItalic,open,TopLeftZoom,659,0,0.0
1.5 Summable Functions and Their Fourier Series/47,Black,notBold,notItalic,open,TopLeftZoom,998,0,0.0
1.6* Gibbs' Phenomenon/54,Black,notBold,notItalic,open,FitWidth,-1
1.7 Miscellaneous Applications/57,Black,notBold,notItalic,open,FitWidth,-1
1.8 Applications to the Partial Differential Equations of One-Dimensional Mathematical Physics/71,Black,notBold,notItalic,open,TopLeftZoom,641,0,0.0
1.9* More General Eigenfunction Expansions/83,Black,notBold,notItalic,open,FitWidth,-1
1.10 Several-Dimensional Fourier Series/92,Black,notBold,notItalic,open,FitWidth,-1
Chapter 2. Fourier Integrals/97,Black,notBold,notItalic,open,FitWidth,-1
2.1 Fourier Integrals/97,Black,notBold,notItalic,open,FitWidth,-1
2.2 Fourier Integrals for C^?_?(R1)/99,Black,notBold,notItalic,open,TopLeftZoom,365,0,0.0
2.3 Fourier Integrals for L²(R1): First Method/102,Black,notBold,notItalic,open,FitWidth,-1
2.4* Fourier Integrals for L²(R1}: Second Method/105,Black,notBold,notItalic,open,FitWidth,-1
2.5* Fourier Integrals for L²(R1): Third Method/108,Black,notBold,notItalic,open,FitWidth,-1
2.6 Fourier Integrals for L²(R1)/112,Black,notBold,notItalic,open,FitWidth,-1
2.7 Miscellaneous Applications/117,Black,notBold,notItalic,open,FitWidth,-1
2.8* Heisenberg's Inequality/127,Black,notBold,notItalic,open,FitWidth,-1
2.9* Band- and Time-Limited Functions/132,Black,notBold,notItalic,open,FitWidth,-1
2.10 Several-Dimensional Fourier Integrals/143,Black,notBold,notItalic,open,FitWidth,-1
2.11 Miscellaneous Applications of Several-Dimensional Fourier Integrals/145,Black,notBold,notItalic,open,FitWidth,-1
Chapter 3. Fourier Integrals and Complex Function Theory/155,Black,notBold,notItalic,open,FitWidth,-1
3.1 A Short Course in Function Theory/155,Black,notBold,notItalic,open,TopLeftZoom,492,0,0.0
3.2 Hardy's Theorem/167,Black,notBold,notItalic,open,FitWidth,-1
3.3 The Paley-Wiener Theorem/169,Black,notBold,notItalic,open,FitWidth,-1
3.4 Hardy Functions/172,Black,notBold,notItalic,open,FitWidth,-1
3.5* Hardy Functions and Filters/181,Black,notBold,notItalic,open,TopLeftZoom,259,0,0.0
3.6* Wiener-Hopf Factorization: Milne's Equation/187,Black,notBold,notItalic,open,FitWidth,-1
3.7* Spitter's Identity/195,Black,notBold,notItalic,open,TopLeftZoom,486,0,0.0
3.8* Hardy Functions in the Disk and Szegö's Theorem/198,Black,notBold,notItalic,open,TopLeftZoom,744,0,0.0
3.9* Polynomial Approximation: The Szász-Müntz Theorem/205,Black,notBold,notItalic,open,TopLeftZoom,639,0,0.0
3.10* The Prime Number Theorem/207,Black,notBold,notItalic,open,TopLeftZoom,555,0,0.0
Chapter 4. Fourier Series and Integrals on Groups/214,Black,notBold,notItalic,open,TopLeftZoom,218,0,0.0
4.1 Groups/214,Black,notBold,notItalic,open,TopLeftZoom,492,0,0.0
4.2 Fourier Series on the Circle/217,Black,notBold,notItalic,open,FitWidth,-1
4.3 Fourier Integrals on the Line/220,Black,notBold,notItalic,open,TopLeftZoom,800,0,0.0
4.4 Finite Commutative Groups/225,Black,notBold,notItalic,open,TopLeftZoom,674,0,0.0
4.5 Fourier Series on a Finite Commutative Group/228,Black,notBold,notItalic,open,TopLeftZoom,800,0,0.0
4.6* Gauss' Law of Quadratic Reciprocity/233,Black,notBold,notItalic,open,TopLeftZoom,695,0,0.0
4.7 Noncommutative Groups/237,Black,notBold,notItalic,open,TopLeftZoom,758,0,0.0
4.8 The Rotation Group/239,Black,notBold,notItalic,open,TopLeftZoom,496,0,0.0
4.9 Three Convolution Algebras/248,Black,notBold,notItalic,open,TopLeftZoom,233,0,0.0
4.10 Homomorphisrns of L¹(K/G/K)/250,Black,notBold,notItalic,open,TopLeftZoom,507,0,0.0
4.11 Spherical Functions Are Eigenfunctions of the Laplacian/253,Black,notBold,notItalic,open,TopLeftZoom,760,0,0.0
4.12 Spherical Functions Are Legendre Polynomials/256,Black,notBold,notItalic,open,TopLeftZoom,550,0,0.0
4.13 Spherical Harmonics/260,Black,notBold,notItalic,open,FitWidth,-1
4.14* Representations of SO(3)/266,Black,notBold,notItalic,open,TopLeftZoom,659,0,0.0
4.15* The Euclidean Motion Group/272,Black,notBold,notItalic,open,FitWidth,-1
4.16* SL(2,R) and the Hyperbolic Plane/284,Black,notBold,notItalic,open,TopLeftZoom,613,0,0.0
Additional Reading/293,Black,notBold,notItalic,open,FitWidth,-1
Bibliography/244,Black,notBold,notItalic,open,FitWidth,-1
Index/300,Black,notBold,notItalic,open,FitWidth,-1