GENERALIZATION OF THE LEFT BERNSTEIN QUASI-INTERPOLANTS

P. Sablonniere introduced the so-called left Bernstein quasi-interpola nt, and proved that the sequence of the approximating polynomials conv erges pointwise in high-order rate to each sufficiently smooth approxi mated function. On the other hand. Z.-C. Wu proved that the sequence o f the norms of the operators is bounded. In this paper, we extract the essence why Sablonniere's operator exhibits good convergence and stab ility properties, and we clarify a sufficient condition for general op erators to have similar properties. Moreover, regarding the family of the general operators, we derive detailed results about the derivative s of the approximating polynomials that estimate their uniform converg ence degree, using a convenient differentiability condition on approxi mated functions. Our results readily imply all the preceding ones. (C) 1998 Academic Press.