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Course, academic year 2019/2020
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Harmonic Analysis 1 - NMAG533
Title in English: Harmonická analýza 1
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:3/1 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
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. (22.02.2019)

Credit and exam, which is oral with a written preparation.

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. (28.06.2017)

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

2) Recall of Topology (final and inicial topology, compact sets via FIP and nets, local compactness, compactification, Alexandrov compactifgication, Tychonov theorem on products of compact sets) and basics on Measure theory (definitions, exaples, inner regular measures, Radon meaure)

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

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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)

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6) Locally compact geroups (definitnion and examples), Haar measure for locally compact groups (existence with a proof, uniqueness without proof, modular function)

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

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),*) anf (L_1(G), . )

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

10) Pontryagin duality (proof)

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

 
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