1l1AB3 | Analysis | Computer Science | S5 | ||||||
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Lessons : 16 h | TD : 20 h | TP : 0 h | Project : 0 h | Total : 36 h | |||||
Co-ordinator : Loick Lhote |
Prerequisite | |
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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. |
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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) |
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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 |
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Acquired skills | |
Modern analysis tools: measures, distributions, Fourier transform General principles of reasonings for analysis: approximation, weak solutions. |
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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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