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El. knyga: Principles of Harmonic Analysis

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  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 21-Jun-2014
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319057927
Kitos knygos pagal šią temą:
Principles of Harmonic AnalysisDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 21-Jun-2014
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319057927
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This book examines the central principles of abelian harmonic analysis: Pontryagin duality, the Plancherel theorem and the Poisson summation formula, as well as their respective generalizations to non-abelian groups, including the Selberg trace formula.

This book offers a complete and streamlined treatment of the central principles of abelian harmonic analysis: Pontryagin duality, the Plancherel theorem and the Poisson summation formula, as well as their respective generalizations to non-abelian groups, including the Selberg trace formula. The principles are then applied to spectral analysis of Heisenberg manifolds and Riemann surfaces. This new edition contains a new chapter on p-adic and adelic groups, as well as a complementary section on direct and projective limits. Many of the supporting proofs have been revised and refined. The book is an excellent resource for graduate students who wish to learn and understand harmonic analysis and for researchers seeking to apply it.
1 Haar Integration
1(36)
1.1 Topological Groups
1(4)
1.2 Locally Compact Groups
5(1)
1.3 Haar Measure
6(8)
1.4 The Modular Function
14(3)
1.5 The Quotient Integral Formula
17(5)
1.6 Convolution
22(3)
1.7 The Fourier Transform
25(1)
1.8 Limits
26(7)
1.9 Exercises
33(4)
2 Banach Algebras
37(24)
2.1 Banach Algebras
37(3)
2.2 The Spectrum σA(a)
40(3)
2.3 Adjoining a Unit
43(2)
2.4 The Gelfand Map
45(3)
2.5 Maximal Ideals
48(1)
2.6 The Gelfand-Naimark Theorem
49(5)
2.7 The Continuous Functional Calculus
54(3)
2.8 Exercises and Notes
57(4)
3 Duality for Abelian Groups
61(24)
3.1 The Dual Group
61(3)
3.2 The Fourier Transform
64(2)
3.3 The C*-Algebra of an LCA-Group
66(3)
3.4 The Plancherel Theorem
69(4)
3.5 Pontryagin Duality
73(4)
3.6 The Poisson Summation Formula
77(3)
3.7 Exercises and Notes
80(5)
4 The Structure of LCA-Groups
85(22)
4.1 Connectedness
85(8)
4.2 The Structure Theorems
93(12)
4.3 Exercises
105(2)
5 Operators on Hilbert Spaces
107(16)
5.1 Functional Calculus
107(4)
5.2 Compact Operators
111(3)
5.3 Hilbert-Schmidt and Trace Class
114(5)
5.4 Exercises
119(4)
6 Representations
123(10)
6.1 Schur's Lemma
123(4)
6.2 Representations of L1(G)
127(3)
6.3 Exercises
130(3)
7 Compact Groups
133(20)
7.1 Finite Dimensional Representations
133(2)
7.2 The Peter-Weyl Theorem
135(7)
7.3 Isotypes
142(2)
7.4 Induced Representations
144(2)
7.5 Representations of SU(2)
146(4)
7.6 Exercises
150(3)
8 Direct Integrals
153(12)
8.1 Von Neumann Algebras
153(1)
8.2 Weak and Strong Topologies
154(1)
8.3 Representations
155(4)
8.4 Hilbert Integrals
159(1)
8.5 The Plancherel Theorem
160(1)
8.6 Exercises
161(4)
9 The Selberg Trace Formula
165(20)
9.1 Cocompact Groups and Lattices
165(2)
9.2 Discreteness of the Spectrum
167(5)
9.3 The Trace Formula
172(5)
9.4 Locally Constant Functions
177(1)
9.5 Lie Groups
177(5)
9.6 Exercises
182(3)
10 The Heisenberg Group
185(10)
10.1 Definition
185(1)
10.2 The Unitary Dual
186(4)
10.3 The Plancherel Theorem for H
190(1)
10.4 The Standard Lattice
190(3)
10.5 Exercises and Notes
193(2)
11 SL2(R)
195(30)
11.1 The Upper Half Plane
195(5)
11.2 The Hecke Algebra
200(8)
11.3 An Explicit Plancherel Theorem
208(1)
11.4 The Trace Formula
209(6)
11.5 Weyl's Asymptotic Law
215(3)
11.6 The Selberg Zeta Function
218(4)
11.7 Exercises and Notes
222(3)
12 Wavelets
225(22)
12.1 First Ideas
225(5)
12.2 Discrete Series Representations
230(5)
12.3 Examples of Wavelet Transforms
235(8)
12.4 Exercises and Notes
243(4)
13 p-Adic Numbers and Adeles
247(22)
13.1 p-Adic Numbers
247(7)
13.2 Haar Measures on p-adic Numbers
254(3)
13.3 Adeles and Ideles
257(8)
13.4 Exercises
265(4)
Appendix A Topology 269(18)
Appendix B Measure and Integration 287(22)
Appendix C Functional Analysis 309(14)
Bibliography 323(4)
Index 327
Anton Deitmar is a professor of Mathematics at the University of Tübingen, Germany. Siegfried Echterhoff is a professor of Mathematics at the University of Münster, Germany.
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