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

Résumé

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 M_n f: M_nf(x) = (n + 1) ∑ⁿ_{k = o}P_nk(x)∝¹₀ P_nk(t) f(t) dt. If the derivative $\text{d}{}^{\mathrm{r}}\text{f}\text{dx}{}^{\mathrm{r}}$ with r ⩾ 0 is continuous on [0, 1], $\text{d}{}^{\mathrm{r}}\text{dx}{}^{\mathrm{r}}\text{M}{}_{\mathrm{n}}\text{f}$ converge uniformly on [0,1] and sup_xϵ[0,1] $\text{¦M}{}_{\mathrm{n}}\text{f(x) − f(x)¦ ⩽ 2ω}{}_{\mathrm{f}}\text{(1/trn)}$ 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).