Fourier Analysis

Module MA4064

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information

MA4064 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

  • Catalogue of non-physics elective courses
Total workloadContact hoursCredits (ECTS)
150 h 45 h 5 CP

Content, Learning Outcome and Preconditions

Content

1. Fourier series. Short review of the classical convergence theorem of Fourier series of Hölder continuous functions. L^2 convergence of Fourier series of L^2 functions and isometry between L^2 and l^2. Regularity and Fourier decay. Selected applications of Fourier series. 2. Fourier transform. Definition on L^1(R^n) and basic properties (inversion formula; behaviour under multiplication, convolution, differentiation). Definition on L^2 and Plancherel's formula. The space of tempered distributions and Fourier calculus on distributions. Periodic arrays of delta functions and Poisson summation. Selected applications of the Fourier transform, e.g. solution of partial differential equations, Heisenberg uncertainty, X-ray crystallography, Shannon sampling and digitalization of acoustic signals, construction of wavelets.

Learning Outcome

After participating in the module, students understand and are able to apply the key mathematical principles of Fourier analysis on euclidean space. They have also obtained some insight into the use of Fourier analysis in contemporary areas of mathematics and the sciences.

Preconditions

MA2003 Measure and Integration

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

ArtSWSTitelDozent(en)Termine
VO 2 Fourier Analysis Friesecke, G. Donnerstag, 10:15–11:45

Learning and Teaching Methods

Vorlesung, Übung, Bearbeitung von Hausaufgaben

Media

Tafel, Übungsblätter

Literature

G. Friesecke, Lectures on Fourier Analysis, Vorlesungsskript (Warwick University, 2007). R. Strichartz, A guide to distribution theory and the Fourier transform (CRC Press, 1994). M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic Press, 1975).

Module Exam

Description of exams and course work

Klausur

Exam Repetition

There is a possibility to take the exam at the end of the semester.

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