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Course, academic year 2023/2024
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Principles of Harmonic Analysis - NMAG533
Title: Principy harmonické analýzy
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Svatopluk Krýsl, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Geometry
Annotation -
Last update: T_MUUK (13.05.2015)
General harmonic analysis generalizes the classical Fourier analysis and the correspondiong analysis of partial differential equations for other groups than the translational R^n. First part of the lecture.
Aim of the course -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (05.01.2017)

Study of harmonic analysis for locally compact groups.

Course completion requirements -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (27.10.2019)

Credit and exam, which is oral with a written preparation. Credit is given by an active presence at exercises (computing at the black-borad) or by computing 15 exercises at home, aliquoat counting.

Literature -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (05.01.2017)

Deitmar, A., Echterhoff, S., Principlesof harmonic analysis

Dixmier, J., C*-algebras and their representations, North-Holland, 1989

Segal, I. E., The group algebra of a locally compact group, Trans. Amer. Math. Soc. 61, 1947, 69-105

Teaching methods -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (13.05.2015)

Lecture and exercise.

Requirements to the exam -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (11.05.2015)

We test the knowledge of definitions, theorems and their application.

Syllabus -
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (14.06.2022)

1) Introduction: Fourier transform and Fourier series, F. transform of Gaussian (by Cauchy theorem)

2) Recall of Topology (final and initial topology, local compactness, compactification, Alexandrov compactification, Tychonov theorem on products of compact sets) and basics on Measure theory (definitions, exapmles, Borel and Radon measures)

3) Compact-open topology, locally uniform convergence on compact sets and Banach--Alaoglu theorem

4) Basics on Banach, Banach-* and C*-algebras (spectrum, resolvent, theorem of Gelfand and Mazur without proof), examples: C(X), B(H), D (disc algebra)

5) Theorem on the Gelfand transform, theorem of Stone-Weierstrass and Gelfand-Naimark (two last without proof)

6) Locally compact groups (definition and examples), Haar measure for locally compact groups (existence with proof, uniqueness without proof), modular functions

7) Basics on representation theory of (topological) groups: Schur lemma (on intertwining homomorphisms), representations of commutative groups, characters of groups

8) L^1(G) with convolution and L^1-norm is Banach *-algebra, group algebra of a finite group, Fourier transform on locally compact groups, F. t. is homomorphism of (L^1(G),*) and (L^1(G), . )

9) Characters. Characters of Z, S^1, R^n. Characters as a locally compact group, Plancherel measure and theorem (without proof)

10) Pontryagin duality (proof)

11) Poisson summation formula on locally compact abelian groups (if time permits transformation rules for theta-functions)

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