Shape preserving approximation by Bernstein-type operators which fix polynomials

In this paper we present a family of sequences of linear Bernstein-type operators, an element of which was recently introduced by J.P. King. As well as showing that they fix certain polynomials and studying some shape preserving and convergence properties, we make a comparison with Bernstein polynomials which shows that for certain usual situations and without too much additional computational cost, the elements of the family represent a good shape preserving approximation process versus that of the referred polynomials. The notion of convexity with respect to a function plays an important role.