A family of Bernstein quasi-interpolants on [0,1]

Suppose that we want to approximate f∈C[0,1] by polynomials inP, using only its values on X_n={i/n, 0≤i≤n}. This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f. But, when n tends to infinity, L_n f does not converge to f in general and the convergence of B_n f to f is very slow. We define a family of operators B _n ^(k) , n≥k, which are intermediate ones between B _n ⁽⁰⁾ =B _n ⁽¹⁾ =B_n and B _n ⁽ⁿ⁾ =L_n, and we study some of their properties. In particular, we prove a Voronovskaja-type theorem which asserts that B _n ^(k) f−f=O(n^−[(k+2)/2]) for f sufficiently regular. Moreover, B _n ^(k) f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.