CLASSES OF FUNCTIONS WITH IMPROVED ESTIMATES IN APPROXIMATION BY THE MAX-PRODUCT BERNSTEIN OPERATOR

摘要

In this paper, we find large classes of positive functions, others than those in [1], having even a Jackson-type estimate, ω1(f;1), in approximation by the nonlinear max-product Bernstein operator. The uniform estimate of the order O[nω1(f;1)2 + ω1(f;1)] is achieved, while near to the endpoints 0 and 1, the better pointwise estimate of the order is obtained. Finally, we prove that besides the preservation of quasi-convexity found in [1], the nonlinear max-product Bernstein operator preserves the quasi-concavity too.