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Analysis

1l1AB3 Analysis Computer Science S5
Lessons : 16 h TD : 20 h TP : 0 h Project : 0 h Total : 36 h
Co-ordinator : Loick Lhote
Prerequisite
Basic math symbols and notions (real/complex number sequences,...)
Course Objectives
Establish the theoretical framework underpinning harmonic analysis (Functions spaces, measures, and distributions).

Get in touch with the general principles of Fourier transform and its applications.

Introduction to functional analysis techniques.
Syllabus
1) Measure and integration

- Abstract measure space: measures, Lebesgue integral, convergence theorem, Fubinit-Tonelli theorem
- Euclidean spaces: Lebesgue measure, continuity/differentiation under integrals, change of variables, fundamental theorem of calculus

2) Espace de fonctions et distributions

- Espaces Lp (p>=1): normes Lp et espaces de Banach, cas p=2, sous-espaces denses, convolution, TF, techniques d'interpolations

- Distributions: fonction tests, espace de Schwartz, semi-norme, distribution, distribution tempérée, exemples de distributions ( distribution associée à une fonction L1loc notamment l'echelon de Heaviside, distributions de Dirac), principe d'extension par dualité linéaire (e.g. dérivées faibles, TF, convolution)
Practical work (TD or TP)
Several applications will be considered during tutorials in particular. Here are a few examples

- Shannon sampling theorem

- Eisenberg uncertainty principle
Acquired skills

Modern analysis tools: measures, distributions, Fourier transform

General principles of reasonings for analysis: approximation, weak solutions.
Bibliography

Rudin, Walter. Real and complex analysis (3rd). New York: McGraw-Hill Inc, 1986.

Tao, Terence, ed. An introduction to measure theory. Vol. 126. AMS Bookstore, 2011.

http://terrytao.wordpress.com/

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