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# Fourier and Laplace Transforms

## Module MA5039

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

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

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2012/3SS 2012

### Basic Information

MA5039 is a semester module in English language at Master’s level which is offered irregular.

This module description is valid from SS 2012 to WS 2018/9.

270 h 90 h 9 CP

### Content, Learning Outcome and Preconditions

#### Content

Fourier Series: Approximation kernels, pointwise convergence, Hilbert-transform, theorem of Bochner.
Fourier Integral: Inversion, Approximation kernels, L2-Theory, sampling theorems.
Gabor-tranform. Laplace-transform.

#### Learning Outcome

At the end of the module, students are able to understand the interplay between functions and their Fourier transform and can use it to obtain optimized approximation of functions.

#### Preconditions

MA1001 Analysis 1, MA1002 Analysis 2, MA3001 Functional Analysis

### Courses, Learning and Teaching Methods and Literature

#### Learning and Teaching Methods

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 animate 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, assignments

#### Literature

P. Butzer, R.Nessel: Fourier Analysis and Approximation. Birkhäuser.
K. Chandrasekharan: Classical Fourier Transforms. Springer.
R. Lasser: Introduction to Fourier Series. Marcel Dekker.

### Module Exam

#### Description of exams and course work

The module examination is based on a 30-minute oral exam. Students are able to understand the interplay between functions and their Fourier transform and can use it to obtain optimized approximations of functions.

#### Exam Repetition

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

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