On preservation of binomial operators

Abstract

Binomial operators are the most important extension to Bernstein operators, defined by $$ \bigl(L^{Q}_{n} f\bigr) (x)=\frac{1}{b_{n}(1)} \sum ^{n}_{k=0}\binom { n}{k } b_{k}(x)b_{n-k}(1-x)f\biggl( \frac{k}{n}\biggr),\quad f\in C[0, 1], $$ ( L n Q f ) ( x ) = 1 b n ( 1 ) ∑ k = 0 n ( n k ) b k ( x ) b n − k ( 1 − x ) f ( k n ) , f ∈ C [ 0 , 1 ] , where $\{b_{n}\}$ { b n } is a sequence of binomial polynomials associated to a delta operator Q. In this paper, we discuss the binomial operators $\{L^{Q}_{n} f\}$ { 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.