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A Generalization of Bernstein–Kantorovič Operators

✍ Scribed by Jesús de la Cal; Ana M Valle

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
125 KB
Volume
252
Category
Article
ISSN
0022-247X

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✦ Synopsis

In this paper, we use a probabilistic setting to introduce a double sequence Ž ² k : . L of linear polynomial operators which includes, as particular cases, the n classical Bernstein operators, the Kantorovic operators, and the operators recently ǐntroduced by Cao. For these operators, we discuss several approximation properties. In particular, we deal with the convergence properties according to the way in which the different parameters vary, and the preservation of global smoothness and classes of functions determined by concave moduli of continuity. A remarkable feature of our approach is that if f is differentiable, the approximation properties of both L ² k : f and its derivatives can be discussed simultaneously. Throughout the n paper, probabilistic methods play an important role.

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## Abstract Let __H__(U) be the space of all analytic functions in the unit disk U, and let co__E__ denote the convex hull of the set __E__ ⊂ ℂ. If __K__ ⊂ __H__(U) then the operator I : __K__ → __H__(U) is said to be an __averaging operator__ if For a function __h__ ∈ __A__ ⊂ __H__(U) we will det