Simultaneous Approximation by Polynomial Projection Operators

Pointwise estimates are obtained for simultaneous approximation of a function f and its derivatives by means of an arbitrary sequence of bounded projection operators with some extra condition (1.3) (we do not require the operators to be linear) which map C_[-1,1] into polynomials of degree n, augmented by the interpolation of f at some points near ±1. The present result essentially improved those in [BaKi3], and several applications are discussed in Section 4.