2024/2025

Harmonic Analysis and Banach Algebras

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 1, 2 module

Open to: students of all HSE University campuses

Instructors: Alexei Yu. Pirkovskii

Language: English

ECTS credits: 3

Contact hours: 30

Course Syllabus

Abstract

Harmonic analysis on locally compact abelian groups is a natural generalization of the classical Fourier analysis usually studied by undergraduate students in mathematics (that is, of the theory of trigonometric Fourier series and of the Fourier transform on the real line). The most elegant approach to harmonic analysis on abelian groups is based on the theory of commutative Banach algebras, which was initiated by Gelfand in the early 1940ies and was further developed by Raikov, Naimark, Shilov and many other brilliant mathematicians. This approach, in particular, yields a relatively simple analytic proof of the Pontryagin duality based on the Plancherel theorem. In this course, we discuss the basics of Banach algebra theory and apply it to constructing the harmonic analysis on a locally compact abelian group. If time permits, some nonabelian groups will also be considered.