Bernstein-type operators which preserve polynomials
In this paper we present the sequence of linear Bernstein-type operators defined for f?C[0,1] f ? C [ 0 , 1 ] by Bn(f???1)?? B n ( f ? ? ? 1 ) ? ? , Bn B n being the classical Bernstein operators and ? ? being any function that is continuously differentiable ? ? times on [0,1] [ 0 , 1 ] , such that ?(0)=0 ? ( 0 ) = 0 , ?(1)=1 ? ( 1 ) = 1 and ??(x)>0 ? ? ( x ) > 0 for x?[0,1] x ? [ 0 , 1 ] . We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with Bn B n . We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve Bn B n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to ? ? plays an important role.
bernstein-type operator positive linear operator voronovskaja-type formula generalized convexity