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Abstract Harmonic Analysis [MA5065] - SS 16

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Abstract harmonic analysis builds on the theory of topological groups that combines the local concept of 'nearness' (given by a topology) with the global concept of 'homomgeneity' (given by a group law). The interplay between these two structures results in a rich theory that finds applications in

This course covers the basic results and techniques of the following topics:

The theory will be accompanied by illustrating examples, and exercises.

(This course is an adaption of last year's course Analysis on Groups.)

News

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Timetable

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Lecture 1   Wednesday,    16:15 - 17:45,   MI 02.10.011  
Lecture 2   Friday,    10:15 - 11:45,   MI 03.08.022  

Exercises   Friday,    12:15 - 13:45   MI 03.08.022  

Exam (25 minutes, oral)  
04.08.2016   10:00 - 11:00    

Repeat Exam  
tba   tba   tba  

Course Material

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Week 1 (25.04. - 29.04.)
Lecture 1: Introduction; background: general topology Lecture 1
Lecture 2: background: uniform spaces; topological groups Lecture 2
Exercises: Some groups; uniform structure of topological groups; Banach spaces; countable groups Sheet 1   Solution

Week 2 (02.05. - 06.05.)
Lecture 3: topological and uniform structure of lcH groups Lecture 3
Lecture 4: topological subgroups and quotients Lecture 4
Exercises: Semidirect product groups; the field of p-adic numbers Sheet 2   Solution

Week 3 (09.05. - 13.05.)
Lecture 5: proper group actions and homogeneous spaces Lecture 5
Lecture 6: functions and measures on locally compact Hausdorff spaces Lecture 6
Exercises: conjugation; point measures on groups Sheet 3   Solution

Week 4 (16.05. - 20.05.)
Lecture 7: the Haar measure of an lcH group Lecture 7
Lecture 8: the Haar measure of an lcH group (ctd.); finiteness properties of the Haar measure Lecture 8
Exercises: some Haar measures; the Haar measure of the shearlet group Sheet 4   Solution

Week 5 (23.05. - 27.05.)
Lecture 9: the modular function; orbital mean operators Lecture 9
Lecture 10: the Weil formula for strongly proper G-spaces Lecture 10
Exercises: modulus of an automorphism; the Haar measure of a semidirect product; the Haar measures of SO(3) and SE(3) Sheet 5   Solution

Week 6 (30.05. - 03.06.)
Lecture 11: proof of the Weil formula; examples Lecture 11
Lecture 12: integration on fundamental domains; the algebras L^1(G) and M(G) Lecture 12
Exercises: A Weil formula for the `ax+b'-group; invariance properties of measures on quotients Sheet 6   Solution

Week 7 (06.06. - 10.06.)
Lecture 13: dual group of an lca group; Fourier transform on L^1(G); Riemann-Lebesgue lemma Lecture 13
Lecture 14: Translation/modulation duality; convolution theorem; F^1(G_hat) is dense in C_0(G_hat) Lecture 14
Exercises: Translation/modulation duality; convolution theorem; the discrete Fourier transform; Heisenberg group and phase space Sheet 7   Solution

Week 8 (13.06. - 17.06.)
Lecture 15: the Fourier transform on L^1(\Q); the ring of adeles Lecture 15
Lecture 16: the Fourier transform on M(G); inverse Fourier transform; positive definite functions Lecture 16
Exercises: the Fourier transform on \R_+, the Mellin transform, and the Riemann zeta function Sheet 8   Solution

Week 9 (20.06. - 24.06.)
Lecture 17: properties of positive definite functions; irreducible unitary representations Lecture 17
Lecture 18: Gelfand-Naimark-Segal construction Lecture 18
Exercises: unitary represenations Sheet 9   Solution

Week 10 (27.06. - 01.07.)
Lecture 19: Bochner's theorem Lecture 19
Lecture 20: Fourier inversion on B(G); Gelfand-Raikov for lca groups Lecture 20
Exercises: discrete and compact groups; Fourier transform of a product; FT on \R and differentiation Sheet 10   Solution

Week 11 (04.07. - 08.07.)
Lecture 21: Pontryagin duality; Fourier inversion on L^1(G); Plancherel's theorem Lecture 21
Lecture 22: the reciprocal group; the Poisson summation formula Lecture 22
Exercises: Shannon sampling on lca groups Sheet 11   Solution

Week 12 (11.07. - 15.07.)
Lecture 23: The Zak transform for lca groups Lecture 23
Lecture 24: The voice transform for affine representations; the continuous wavelet transform Lecture 24
Exercises: no exercises  

Literature

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Harmonic Analysis
Jüstel   Abstract Harmonic Analysis    lecture notes    2016       
Folland   A Course in Abstract Harmonic Analysis    CRC Press    1995       
Reiter, Stegeman   Classical Harmonic Analysis and Locally Compact Groups    Oxford University Press    2000       
Hewitt, Ross   Abstract Harmonic Analysis (Volume I and II)    Springer    1963, 1970       

TopologySorted descending
Kelley   General Topology    Van Nostrand    1955       
Bourbaki   General Topology    Springer    1989       

People

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  Name eMail Room Consultation
Lecturer   Dominik Jüstel  juestelma.tum.de   MI 03.06.021   tba