# Fallstudien der Mathematischen Modellbildung: Teil 1 (MA 2902) - WS 15/16

Aktuelles Inhalt Slides Übungsblätter Matlab Codes LiteraturDozent | Massimo Fornasier | massimo.fornasierma.tum.de | MI 02.10.058 |

Übungsleitung | Mattia Bongini | mattia.bonginima.tum.de | MI 02.10.040 |

Vorlesungstermine | Di. 12:15 - 13:45 | 00.06.011, MI Hörsaal 3 (5606.EG.011) | |

Mi. 16:00 - 17:30 | CH 22210, Ivar-Ugi-Hörsaal (5402.01.221K) | ||

Übungstermin | Mo. 16.15 - 17.45 | 00.06.011, MI Hörsaal 3 (5606.EG.011) |

## Aktuelles

nach oben ***02.12.2014**: The first exercise session is confirmed to be held on 12.10.15 in Room 00.06.011. We ask the students to practice with Hilbert spaces on the set of exercises we propose. An excellent reference is W. Rudin, "Real and complex analysis", 1987. *

**14.12.2014**: The second exercise sheet is available. As a reference, consult the book [5].

## Inhalt der Vorlesung

nach oben Can one of the most important Italian Renaissance frescoes reduced in hundreds of thousand fragmentsby a bombing during the Second World War bere-composed after more than 60 years from its damage? Can we

reconstruct the missing parts and can we say something about their original color?

Our lectures on Fallstudien der Modellbildung starts by exemplifying, hopefully effectively by taking advantage of the seduction of art,

how mathematics today can be applied in real-life problems which were considered unsolvable only few years ago,

see reference [1].

We shall continue our lectures digging into the world of digitalization, i.e. how images, sounds, signals, opinions,

emotions, etc. can be made numbers and then elaborated via mathematical algorithms. For that we will need to learn

some of the fundamentals of harmonic analysis, see reference [3], in particular the Fourier theorem in Hilbert spaces

and its concrete application to define Fourier series, transforms, and the algorithm of the Fast Fourier Transform (FFT),

see [Chapter 1, 2] and [3]. This will introduces us to the problem of estimating how good is the approximation in

computing a Fourier Transform of a function from its samples, and the Shannon sampling theory, see [Chapter 2, 2] and [3].

To address the analysis of signals in their time-frequency nature, we shall explore the tools provided by Gabor

analysis, in particular the so-called Gabor transform and its discretization via frames, see [Chapter 3, 2] and [4,5,6].

We conclude the lectures again by returning to the beginning, and by analysing in details the mathematics behind the fresco

restoration problem, as our main inspirational Fallstudium.

**In these lectures we follow very closely the Skriptum [2], which collects in short several results from other texts, in particular**

[3,4,5,6]. As the Skriptum is currently available only in Italian we shall give at the end of each lecture a synthesis in the form

of Slides (Folien) in English, which will be posted online in PDF. The course and the exercises in this part of Fallstudien will

be held in English.

[3,4,5,6]. As the Skriptum is currently available only in Italian we shall give at the end of each lecture a synthesis in the form

of Slides (Folien) in English, which will be posted online in PDF. The course and the exercises in this part of Fallstudien will

be held in English.

## Slides

nach oben Introduction lecture, Oct. 12, 2015Slides of the lecture 1, Oct. 15, 2015

Slides of the lecture 2, Oct. 20, 2015

Slides of the lecture 3, Oct. 21, 2015

Slides of the lecture 4, Oct. 27, 2015

Slides of the lecture 5, Oct. 29, 2015

Slides of the lecture 6, Nov. 3, 2015

Slides of the lecture 7, Nov. 4, 2015

Slides of the lecture 8, Nov. 10 , 2015

Mathematica Notebook

## Übungsblätter

nach oben Übungsblatt 1 Übungsblatt 2 Übungsblatt 3 Übungsblatt 4 Übungsblatt 5## Matlab Codes

Discrete Fourier Transform Fast Fourier Transform nach oben## Themen

Themen Part 1 For more info, see the webpage of the exam. nach oben## Literatur

nach oben 1. M. Fornasier, Mathematics enters the picture, Proceedings of the conference Mathknow 2008 [ .pdf ]2. M. Fornasier Introduzione all'analisi armonica numerica (Italian), Lecture notes, 2007 112 pp. [ .pdf ]

3. D. W. Kammler, A First Course in Fourier Analysis, Prentice Hall, Upper Saddle River, New Jersey 07458, 2000. [ ref ]

4. C. Heil, A Basis Theory Primer, Birkhaeuser, 1998. [ .pdf ]

5. O. Christensen, An Introduction to Frames and Riesz Bases, Birkhaeuser, 2003.

6. H. G. Feichtinger, F. Luef, T. Werther, A Guided Tour from Linear Algebra to the Foundations of Gabor Analysis, Univ. of Vienna, August 2005 [ .pdf ]

-- MattiaBongini - 01 Oct 2015