# Coursebooks 2016-2017

Ruppen Hans-Jörg

English

#### Remarque

Advanced Analysis I and Advanced Analysis II must be taken together as a whole

#### Summary

Getting access to the use of Banach spaces, Hilbert spaces, Fourier series, Fourier tansforms and distributions.

#### Content

1. Inner product spaces and Hilbert spaces
2. L2 spaces
3. Orthonormal sets in Hilbert spaces: Fourier coefficients, Bessel inequality and equality
4. Periodic signals and Fourier series
5. Fourier Transform in L1 and in L2
6. Distrribution spaces
7. Tempered distributions and Fourier transform

#### Keywords

inner product spaces, Hilbert spaces, Lp spaces, orthonoral sets, Fourier coefficients, Fourier transform, distributions, tempered distributions, peridoic signals, Dirac comb, sampling of a signal

#### Learning Outcomes

By the end of the course, the student must be able to:
• Explain the main concepts and propositions presented in the lecture
• Detect the main properties (as Banach, Hilbert, norm, inner product) in examples
• Exploit the main propositions in concrete examples
• Formalize the main tools used for signals (sampling,...)
• Theorize the environment in which Fourier analysis is permormed

#### Transversal skills

• Assess one's own level of skill acquisition, and plan their on-going learning goals.
• Continue to work through difficulties or initial failure to find optimal solutions.
• Communicate effectively, being understood, including across different languages and cultures.

#### Teaching methods

Ex cathedra lecture and exercises in the classroom

#### Expected student activities

Understanding the mathematical language necessary for a deep understanding of signals and their transforms, of the Lebesgue spaces and the distribution spaces

Oral exam

#### Resources

##### Bibliography

C. Gasquet, P. Witomski: Fourier Analysis and Applications, Springer, ISBN 0-387-98485-2

W. Kammler: A First Course in Fourier AnalysisDavid, Online ISBN: 9780511619700 Hardback ISBN: 9780521883405 Paperback ISBN: 9780521709798

##### Notes/Handbook

Lecture notes: Advanced Analysis II by Hans-Jörg Ruppen (Librairie La Fontaine)

Diploma

### Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14   BS260
14-15
15-16   BS260
16-17
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German