I Fourier Analysis
1 Fourier Series
1.1 Periodic Functions
1.2 Exponentials
1.3 The Bessel Inequality
1.4 Convergence in the L2-Norm
1.5 Uniform Convergence of Fourier Series
1.6 Periodic Functions Revisited
1.7 Exercises
2 Hilbert Spaces
2.1 Pre-Hilbert and Hilbert Spaces
2.2 2-Spaces
2.3 Orthonormal Bases and Completion
2.4 Fourier Series Revisited
2.5 Exercises
3 The Fourier Transform
3.1 Convergence Theorems
3.2 Convolution
3.3 The Transform
3.4 The Inversion Formula
3.5 Plancherel's Theorem
3.6 The Poisson Summation Formula
3.7 Theta Series
3.8 Exercises
4 Distributions
4.1 Definition
4.2 The Derivative of a Distribution
4.3 Tempered Distributions
4.4 Fourier Transform
4.5 Exercises
II LCA Groups
5 Finite Abelian Groups
5.1 The Dual Group
5.2 The Fourier Transform
5.3 Convolution
5.4 Exercises
6 LCA Groups
6.1. Metric Spaces and Topology
6.2 Completion
6.3 LCA Groups
6.4 Exercises
7 The Dual Group
7.1 The Dual as LCA Group
7.2 PontryaginDuality
7.3 Exercises
8 Plancherel Theorem
8.1 Haar Integration
8.2 Fubini's Theorem
8.3 Convolution
8.4 Plancherel's Theorem
8.5 Exercises
III Noncommutative Groups
9 Matrix Groups
9.1 GLn(C) and U(n)
9.2 Representations
9.3 The Exponential
9.4 Exercises
10 The Representations of SU(2)
10.1 The Lie Algebra
10.2 The Representations
10.3 Exercises
11 The Peter-Weyl Theorem
11.1 Decomposition of Representations
11.2 The Representation on Hom(Vr,VT)
11.3 The Peter-Weyl Theorem
11.4 AReformulation
11.5 Exercises
12 The Heisenberg Group
12.1 Definition
12.2 The Unitary Dual
12.3 Hilbert-Schmidt Operators
12.4 The Plancherel Theorem for H
12.5 AReformulation
12.6 Exercises
A TheRiemannZetaFunction
B Haar Integration
Bibiliography
Index
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