On preservation of binomial operators

Binomial operators are the most important extension to Bernstein operators, defined by (L(n)(Q)f)(x) = 1/b(n)(1) Sigma(n)(k=0)(n k)b(k)(x)b(n-k)(1 - x)f(k/n), f is an element of C[0,1], where {b(n)} is a sequence of binomial polynomials associated to a delta operator Q. In this paper, we discuss the binomial operators {L(n)(Q)f} preservation such as smoothness and semi-additivity by the aid of binary representation of the operators, and present several illustrative examples. The results obtained in this paper generalize what are known as the corresponding Bernstein operators.