On preservation of binomial operators

Abstract

Abstract Binomial operators are the most important extension to Bernstein operators, defined by ( 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 ] , $$ \bigl(L^{Q}_{n} f\bigr) (x)=\frac{1}{b_{n}(1)} \sum ^{n}_{k=0}\binom { n}{k } b_{k}(x)b_{n-k}(1-x)f\biggl( \frac{k}{n}\biggr),\quad f\in C[0, 1], $$ 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 n Q f } $\{L^{Q}_{n} f\}$ 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.

Keywords

• Bernstein operators
• Delta operators
• Binomial operators
• Binary representation
• Preservation
• Smoothness