M470 - Abstract Harmonic Analysis

Course No: 
Topological Groups: Basic properties of topological groups, subgroups, quotient groups. Examples of various matrix groups. Connected groups.
Haar measure: Discussion of Haar measure without proof on $\mathbb R$, $\mathbb T$, $\mathbb Z$ and simple matrix groups, Convolution, the Banach algebra $L_1(G)$ and convolution with special emphasis on $L_1(\mathbb R)$, $L_1(\mathbb T)$ and $L_1(\mathbb Z)$.
Basic Representation Theory: Unitary representation of groups, Examples and General properties, The representations of Group and Group algebras, C* -algebra of a group, GNS construction, Positive definite functions, Schur’s Lemma. Abelian Groups: Fourier transform and its properties, Approximate identities in $L_1(G)$, Classical Kernels on $\mathbb R$, The Fourier inversion Theorem, Plancherel theorem on $\mathbb R$, Plancherel measure on $\mathbb R$, $\mathbb T$, $\mathbb Z$. Dual Group of an Abelian Group: The Dual group of a locally compact abelian group, Computation of dual groups for $\mathbb R$, $\mathbb T$, $\mathbb Z$, Pontryagin’s Duality theorem.
Reference Books: 
  1. G. B. Folland, “A Course in Abstract Harmonic Analysis”, CRC Press, 2000.
  2. H. Helson, “Harmonic Analysis”, Texts and Readings in Mathematics, Hindustan Book Agency, 2010.
  3. Y. Katznelson, “An Introduction to Harmonic Analysis”, Cambridge University Press, 2004.
  4. L. H. Loomis, “An Introduction to Abstract Harmonic Analysis”, Dover Publication, 2011.
  5. E. Hewitt, K. A. Ross, “Abstract Harmonic Analysis Vol. I”, Springer-Verlag, 1979.
  6. W. Rudin, “Real and Complex Analysis”, Tata McGraw-Hill, 2013.

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