## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-65-14-M-7

## Module Identification

Module Number Module Name CP (Effort)
MAT-65-14-M-7 Distributions and Wavelets 4.5 CP (135 h)

## Basedata

CP, Effort 4.5 CP = 135 h 1 Sem. irreg. [7] Master (Advanced) [EN] English Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) [MAT-SPAS] Analysis and Stochastics [MAT-88.105-SG] M.Sc. Mathematics [AUSL] Phase-out period

## Notice

This module is part of the module [MAT-65-12-M-7].

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V MAT-65-14-K-7
Distributions and Wavelets
P 28 h 107 h - - PL1 4.5 irreg.
• About [MAT-65-14-K-7]: Title: "Distributions and Wavelets"; Presence-Time: 28 h; Self-Study: 107 h

## Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: irregular (by arrangement)
• Examination number: 86198 ("Distributions and Wavelets")

## Contents

• test functions and distributions,
• operations on distributions (translation, dilation, differentiation, folding),
• Schwartz functions and tempered distributions,
• Fourier transformations of tempered distributions,
• wavelets and wavelet frames.

## Competencies / intended learning achievements

Upon successful completion of this module, the students have studied and understand the basic concepts of distribution theory. They have deepened their knowledge of wavelets and learnt about applications through examples. They are able to name the essential propositions of the lecture as well as to classify and to explain the connections.

They understand the proofs presented in the lecture and are able to comprehend and explain them. In particular, they are able to outline the conditions and assumptions that are necessary for the validity of the statements.

## Literature

• W. Walter: Einführung in die Theorie der Distributionen,
• I. M. Gelfang, G. E. Schilow: Verallgemeinerte Funktionen I,
• W. Rudin: Functional Analysis,
• R. Strichartz: A Guide to Distribution Theory and Fourier Transform,
• G. B. Folland: Fourier Analysis and its Applications,
• I. Daubechies: Ten Lectures on Wavelets,
• S. Mallat: A Wavelet Tour of Signal Processing.

## Requirements for attendance of the module (informal)

Knowledge in the field of image processing (e.g. from the module [MAT-65-11-M-7]) is helpful, but not necessarily required.

None

## References to Module / Module Number [MAT-65-14-M-7]

Module-Pool Name
[MAT-65-MPOOL-7] Specialisation Image Processing and Data Analysis (M.Sc.)
[MAT-AM-MPOOL-7] Applied Mathematics (Advanced Modules M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)