Structure Properties for Binomial Operators

Abstract

In this paper, we discuss binomial operators structure properties, such as moments representation, derivatives representation, and binary representation and introduce some applications in preservation.

On preservation of binomial operators

“…It is well known that the behavior of an operator strongly depends on its structure. From [40] we can find binomial operators have the following so-called binary representation similar to Bernstein operators.…”

“…Before 1970, Mullin and Rota in [26] introduced a simpler and more convenient operator method, umbral calculus, which is a linear-algebraic theory used to study certain types of polynomial functions that play an important role in applied mathematics [3,18]. Preservation of operators belong to theoretical basis of CAGD, involving preservation of smoothness, preservation of shape, one-side approximation, variation diminishing, the best constants in approximation of preservation and so on, on which there has been considerable research in [1,3,8,16,18,20,40]. Here we would also like to mention Refs.…”

“…. ,the corresponding binomial operators L ∇n are not well defined, in this case being replaced by (see[40]) nx)b n-k (nnx)f k n , f ∈ C[0, 1].Obviously, it is not positive, therefore Theorems 3.1-3.4 as well as their corollaries are no longer true for the binomial operators L ∇ n . If the delta operator Q = T = ln(I + D) (Touchard operator), then t n (x) = e -x…”