Abstract

Binomial operators are the most important extension to Bernstein operators, defined by (LnQf)(x)=1bn(1)∑k=0n(nk)bk(x)bn−k(1−x)f(kn),f∈C[0,1],\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(L^{Q}_{n} f\\bigr) (x)=\\frac{1}{b_{n}(1)} \\sum ^{n}_{k=0}\\binom { n}{k } b_{k}(x)b_{n-k}(1-x)f\\biggl( \\frac{k}{n}\\biggr),\\quad f\\in C[0, 1], $$\\end{document} where {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} 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.

Highlights

  • Bernstein operators, known as Bernstein polynomials, are typical positive linear operators, defined as follows: n (Bnf )(x) =n xk(1 – x)n–kf k, k n n = 1, 2, . . . , k=0 which were first introduced by Bernstein in [6], and more detailed discussions were given by Lorentz in [19]

  • Thanks to its simple and graceful form, as well as the favorable properties of approximation and preservation, Bernstein operators have attracted a good deal of attention, with hundreds of related research publications [2, 7, 9, 11,12,13, 17, 19, 28, 29, 38]

  • Due to the properties of approximation and preservation, Bernstein operators are applied to CAGD and IM

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Summary

Introduction

K bk(x)bn–k(y), k=0 and Q is a delta operator uniquely determined by the sequence of binomial polynomials {bn}. In the present paper we show the binomial operators preserving smoothness and semi-additivity, which appear in the third section. We set out for some well-known results on preservation of smoothness for Bernstein operators as follows ([20], Theorem A).

Results
Conclusion
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