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
 signal and image analysis (where topological groups appear as structure of the signal space or as group of transformations of images),
 physics (where topological groups appear as symmetry groups of equations or theories),
 pure maths (where topological groups play a central role in number theory, as well as algebraic geometry, and topology).
This course covers the basic results and techniques of the following topics:
 integration on groups: existence and uniqueness of a canonical measure (Haar measure), and its integration theory,
 Fourier analysis on abelian groups: the Fourier transform on abelian groups, its basic properties, and some classic theorems, e.g. inversion theorem, Plancherel, Pontryagin duality, Poisson summation formula,
 Further topics: e.g. distributions on groups, the Heisenberg group, the voice transform, maybe compact groups.
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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 01.06.2015: Due to the small number of participants, there will be an oral exam instead of the previously announced written exam.
 01.06.2015: The time of the Wednesday lecture has been changed to 16:15  17:45.
 02.05.2016: The new dates of lectures and exercises are: Lectures: Wednesday, 16:00  17:30, 02.10.011, Friday, 10:15  11:45, 03.08.022, Exercises: Friday, 12:15  13:45, 03.08.022.
 29.04.2016: There will be no lecture on Monday, 02.05.2016, due to the unresolved problems with the dates of lectures and exercises.
 27.04.2016: In the Literature section, the (continuously updated) lecture notes can be found as a single pdffile.
 26.04.2016: The dates of the course are put up for discussion in a Doodle ^{}.
 21.03.2016: Registration for the course is now open in TUMonline. Note that the course will begin in the third week of the semester (25.04.16).
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 

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 padic 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 Gspaces 
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); RiemannLebesgue 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: 
GelfandNaimarkSegal 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); GelfandRaikov 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 
Folland 
A Course in Abstract Harmonic Analysis 
CRC Press 
1995 


Hewitt, Ross 
Abstract Harmonic Analysis (Volume I and II) 
Springer 
1963, 1970 


Jüstel 
Abstract Harmonic Analysis 
lecture notes 
2016 


Reiter, Stegeman 
Classical Harmonic Analysis and Locally Compact Groups 
Oxford University Press 
2000 


Topology 
Kelley 
General Topology 
Van Nostrand 
1955 


Bourbaki 
General Topology 
Springer 
1989 


People
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