A Voronovskaja type formula for Soardi’s operators

Abstract

AbstractIn 1991 Soardi introduced a sequence of positive linear operators $$\beta _{n}$$ β n associating to each function $$f\in C\left[ 0,1\right] $$ f ∈ C 0 , 1 a polynomial function which is closely related to the Bernstein polynomials on $$\left[ -1,+1\right] $$ - 1 , + 1 . One of the authors already studied the operators $$\beta _{n}$$ β n in several papers. This paper is devoted to other properties of Soardi’s operators. We introduce a version $${\tilde{\beta }}_{n}$$ β ~ n which can be expressed in terms of the classical Bernstein operators and present the relations between $$\beta _{n}$$ β n and $${\tilde{\beta }}_{n}$$ β ~ n . We derive Voronovskaja-type results for both $$\beta _{n}$$ β n and $${\tilde{\beta }}_{n}$$ β ~ n . Furthermore, rates of convergence for $${\tilde{\beta }}_{n}$$ β ~ n , respectively $$\beta _{n}$$ β n , are estimated. Finally, we study the first and second moments of $$\beta _{n}$$ β n .