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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.

Module version of SS 2012 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

available module versions
SS 2012WS 2011/2

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

TypeSWSTitleLecturer(s)Dates
VO 2 Fourier Analysis [MA4064] Friesecke, G. Thu, 10:15–11:45, MI 00.06.011
UE 1 Fourier Analysis (Exercise Session) [MA4064] Friesecke, G. Tsipenyuk, A. singular or moved dates
and dates in groups

Learning and Teaching Methods

lecture, exercise module, assignments
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

Media

blackboard

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

The module examination is based on a written exam (60 minutes).

Exam Repetition

The exam may be repeated at the end of the semester.

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