Syllabus
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)
Annotation
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.