Pointwise Weighted Approximation of Functions with Endpoint Singularities by Combinations of Bernstein Operators

Approximation properties of Bernstein operators have been studied very well (see [2], [3], [5]-[8], [12]-[14], for example). In order to approximate the functions with singularities, Della Vecchia et al. [3] and Yu-Zhao [12] introduced some kinds of modified Bernstein operators. Throughout the paper, C denotes a positive constant independent of n and x, which may be different in different cases. Let w(x) = x(1− x), α, β > 0, α+ β > 0, 0 6 x 6 1.