Elements of Harmonic Analysis

What it is about

You will learn the fundamentals of harmonic analysis for different groups: finite groups, the torus and the real line. Topics include Fourier analysis in more abstract settings, convolution operations, representation theory of finite type, as well as central theorems such as Plancherel, inversion, Bochner and duality theorems; at the end you will be able to understand classical results of Fourier analysis in the context of locally compact (abelian) groups and to give elementary proofs.

What you will be able to do

• Understanding of fundamental concepts of harmonic analysis
• Application of Fourier analysis to finite groups, torus and R
• Knowledge of Haar measure and topological groups
• Familiarity with convolution, positive definite functions and basic representation theory elements
• Comprehension and application of Plancherel, inversion, Bochner and duality theorems
• Insight into principles such as the uncertainty principle in the harmonic analysis context

What the module consists of

• VorlesungDelivery of theoretical foundations on the board
• ÜbungWorking on theoretical problems for deepening understanding, partly in team work, with supervision by the instructor

Teaching method

• Tafelvorlesungfor step-by-step and detailed presentation of the theory
• Übungssitzungen mit Aufgabenfor deepening the material through independent and supervised practice, fostering understanding