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

Resume We study here a new kind of modified Bernstein polynomial operators on L 1 (0, 1) introduced by J. L. Durrmeyer in [4]. We define for f integrable on [0, 1] the modified Bernstein polynomial M n f : M n f ( x ) = ( n + 1) ∑ n k = o P nk ( x )∝ 1 0 P nk ( t ) f ( t ) dt . If the derivative d r f dx r with r ⩾ 0 is continuous on [0, 1], d r dx r M n f converge uniformly on [0,1] and sup xϵ[0,1] ¦M n f(x) − f(x)¦ ⩽ 2ω f (1/trn) if ω f is the modulus of continuity of f. If f is in Sobolev space W l , p (0, 1) with l ⩾ 0, p ⩾ 1, M n f converge to f in w l , p (0, 1).