|
|
|
||
Last update: T_MUUK (13.05.2015)
|
|
||
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (05.01.2017)
Study of harmonic analysis for locally compact groups. |
|
||
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. |
|
||
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
|
|
||
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (13.05.2015)
Lecture and exercise. |
|
||
Last update: doc. RNDr. Svatopluk Krýsl, Ph.D. (11.05.2015)
We test the knowledge of definitions, theorems and their application. |
|
||
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) |