Sur l'approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifies

RésuméWe 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 fdxr with r ⩾ 0 is continuous on [0, 1], drdxrMn f converge uniformly on [0,1] and supxϵ[0,1] ¦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)