Analyse numérique et équations différentielles by Demailly Jean-Pierre

By Demailly Jean-Pierre

Demailly J.-P. examine numerique et equations differentielles (EDP Sciences, 2006)(ISBN 286883891X)(fr)(345s)_MN_

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Lemme – Pour tout n ∈ N∗ , max (nδn (x), (n + 1)δn+1 (x)) ≥ 1 min(1 + x, 1 − x) > 0. 2 39 II – Approximation polynomiale des fonctions num´ eriques Il existe en effet des indices j, k tels que δn (x) = |x − xj,n | = x − −1+j· 2 n δn+1 (x) = |x − xk,n | = x − −1+k· 2 n+1 , . On obtient donc max (nδn (x), (n + 1)δn+1 (x)) ≥ 1 2 1 ≥ 2 1 ≥ 2 1 = 2 ≥ 1 (nδn (x) + (n + 1)δn+1 (x)) 2 |n(x + 1) − 2j| + |(n + 1)(x + 1) − 2k| |diff´erence| = 1 |x + 1 − 2k + 2j| 2 distance (x, entiers impairs dans Z) min(|x − 1|, |x + 1|).

Calculer g(θ1 ) − nπ et g(0) − nπ. En d´eduire qu’il existe θ2 v´erifiant 0 < θ2 < θ1 et g(θ2 ) = nπ. (β) Montrer qu’il existe une suite strictement d´ecroissante θk de [0, π] telle que g(θk ) = (n − k + 2)π pour k = 1, . . n + 2. (γ) On note ϕn (x) = Re ei(n+1)θ 1−ce−iθ 1−ceiθ avec θ = Arc cos x, x ∈ [−1, 1]. Calculer ϕn . Montrer que ϕn ´equioscille sur n + 2 points de [−1, 1]. +∞ (c) (α) Montrer que la s´erie − 12 + k=0 ck tk (x) converge uniform´ement sur [−1, 1] vers une fonction fc (x) que l’on explicitera.

Th´ eor` eme 3 – Pour tout poids w sur ]a, b[, le polynˆome pn poss`ede n z´eros distincts dans l’intervalle ]a, b[. D´ emonstration. Soient x1 , . . , xk les z´eros distincts de pn contenus dans ]a, b[ et m1 , . . , mk leurs multiplicit´es respectives. On a m1 + . . + mk ≤ deg pn = n. Posons εi = 0 si mi est pair, εi = 1 si mi est impair, et k q(x) = (x − xi )εi , deg q ≤ k ≤ n. i=1 Le polynˆ ome pn q admet dans ]a, b[ les z´eros xi avec multiplicit´e paire mi + εi , donc pn q est de signe constant dans ]a, b[ \ {x1 , .

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