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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.
Last update: T_MUUK (13.05.2015)
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Teach basics of harmonic analysis, mostly the Fourier transform on locally compact groups and its generalization to Banach *-algebras. Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (28.07.2024)
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Credit and exam, which is oral with a written preparation. Credit is given by an active presence at exercises (computing at the black-board) or by computing at most 10 exercises at home. Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (12.07.2024)
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Deitmar, A., Echterhoff, S., Principles of 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: Krýsl Svatopluk, doc. RNDr., Ph.D. (02.06.2024)
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Lecture and exercise. Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (13.05.2015)
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We test the knowledge of definitions, theorems and their application. Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (11.05.2015)
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1) Introduction Heat equation (for different function spaces), Fourier transform and Fourier series, Fourier transform of the Gaussian function by Cauchy theorem
2) Recall of definitions and theorems in topology and measure theory Final and initial topology, local compactness, Alexandrov compactification, Tychonov theorem on products of compact sets, Banach--Alaoglu theorem. Borel and Radon measures
4) Basics on Banach-* and C*-algebras Spectrum, resolvent, theorem of Gelfand and Mazur, examples: C(X), B(H), D (disc algebra), theorem on the Gelfand transform and theorems of Stone-Weierstrass (without proof) and Gelfand--Naimark
5) Locally compact groups Definition and examples, Haar measure for locally compact groups (existence with proof), modular factors, p-adic groups
6) Basics on representation theory of (topological) groups Schur lemma (on intertwining-equivariant maps), representations of commutative groups, characters of groups
7) L^1(G) L^1(G) is a Banach *-algebra. Group algebra of a finite group, Fourier transform on locally compact groups, Fourier transform is homomorphism of the semigroups (L^1(G),*) and (L^1(G),.)
8) Characters Characters of Z, S^1, R^n. Characters as a locally compact group, Plancherel measure and theorem (without proof)
9) Pontryagin duality (partial proof)
10) Application Poisson summation formula on R^n (or locally compact commutative groups) and transformation rules for theta-functions. Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (12.07.2024)
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