5-08 - AMPHI 5 TRANSFORME DE FOURIER 1 Principaux rsultats...

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AMPHI 5 TRANSFORMÉE DE FOURIER 1
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Principaux résultats d’intégration Théorème de convergence dominée Ensembles de mesure nulle (ou négligeables ) et th. de Borel-Cantelli. Intégration des fonctions positive et théorème de convergence monotone. Complétude des espaces L 1 (X) et L 2 (X) , si X R m . X ouvert = Esc(X) , C c (X) et C c (X) denses dans L 1 (X) et L 2 (X) . Théorème de Fubini et formule du changement de variable. Continuité et dérivabilité d’une fonction définie par une intégrale. 2
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Comparaison des différents modes de convergence X R m , convergence de f n vers f sur X . Convergence uniforme = convergence simple = convergence p.p. Convergence p.p. + domination = convergence dans L 1 (X) ou L 2 (X) (th. de CV dominée). Convergence dans L 1 (X) ou L 2 (X) = sous-suites convergeant p.p. Convergence dans L 2 (X) = convergence dans L 1 (X) si λ (X) < + (Cauchy-Schwarz). 3
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Transformée de Fourier dans L 1 ( R m ) Cadre naturel L 2 ( R m ) ou distributions (l’an prochain).
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