## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Course MAT-65-12-K-7

## Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U) 9.0 CP 186 h
4 V Lecture 56 h
2 U Exercise class (in small groups) 28 h
(4V+2U) 9.0 CP 84 h 186 h

## Basedata

SWS 4V+2U 9.0 CP = 270 h 1 Sem. irreg. [7] Master (Advanced) [EN] English Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) [MAT-SPAS] Analysis and Stochastics [AUSL] Phase-out period

## Notice

This course was offered for the last time in SS 2019.

Students who have already used the course [MAT-65-10-K-4] for one of the specialization modules of the Bachelor's programme in Mathematics can use the second part of the course as module [MAT-65-14-M-7] "Distributions and Wavelets" for their master's studies.

## Contents

• Fourier series (Fourier coefficients and Fourier series, convolution of periodic functions, pointwise and uniform convergence of Fourier series, Gibbs phenomenon),
• Fourier transform (Fourier transform in L1, Fourier transform in L2, Poisson’s summation formula and Shannon’s sampling theorem, Heisenberg’s uncertainty principle, Windowed Fourier transform),
• discrete Fourier transform (approximation of Fourier coefficients and aliasing formula, Fourier matrix and discrete Fourier transform, circulant matrices, Kronecker products and stride permutations, discrete trigonometric transforms),
• fast Fourier transform (Radix-2 algorithm, sparse Fourier transform, Fourier transform for non equispaced data),
• Prony’s method for the reconstruction of structured functions (Prony method, recovery of exponential sums),
• distributions (test functions and distributions, Schwartz spaces and tempered distributions, Fourier transform of tempered distributions),
• wavelets and wavelet frames (continuous wavelet transform, wavelets frames, Haar wavelets, multiresolution analysis).

## Literature

• W. Walter: Einführung in die Theorie der Distributionen,
• 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.

## Materials

Further literature will be announced in the lecture; Exercise material is provided.

## Requirements for attendance (informal)

Knowledge from the courses [MAT-12-28-K-3] and [MAT-14-11-K-3] as well as basic knowledge in the field of image processing are helpful, but not necessarily required.

None

## References to Course [MAT-65-12-K-7]

Module Name Context
[MAT-65-12-M-7] Fourier Analysis with Applications in Image Processing P: Obligatory 4V+2U, 9.0 LP, AUSL