Sur l'approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifies |
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Authors: | Marie Madeleine Derriennic |
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Institution: | Laboratoire d''Analyse Numérique, Institut National des Sciences Appliquees 20, avenue des Buttes de Coësmes, 35043 Rennes Cédex, France |
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Abstract: | We study here a new kind of modified Bernstein polynomial operators on L1(0, 1) introduced by J. L. Durrmeyer in 4]. We define for f integrable on 0, 1] the modified Bernstein polynomial Mn f: Mnf(x) = (n + 1) ∑nk = oPnk(x)∝10 Pnk(t) f(t) dt. If the derivative dr f/dxr with r 0 is continuous on 0, 1], dr/dxrMn f converge uniformly on 0,1] and supxε0,1] ¦Mn f(x) − f(x)¦ 2ωf(1/trn) if ωf is the modulus of continuity of f. If f is in Sobolev space Wl,p(0, 1) with l 0, p 1, Mn f converge to f in wl,p(0, 1). |
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