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Original Articles

On the rates of convergence of Bernstein–Chlodovsky polynomials and their Bézier-type variants

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Pages 403-416 | Received 21 May 2009, Accepted 07 Oct 2009, Published online: 08 Apr 2010
 

Abstract

In this article, we consider the Chlodovsky polynomials C n f and their Bézier variants C n f, with α > 0, for locally bounded functions f on the interval [0, ∞). Using the Chanturiya modulus of variation we give estimates for the rates of convergence of C n f (x) and C n f (x) at those points x > 0 at which the one-sided limits f (x+), f (x−) exist. The recent results of Karsli and Ibiki [H. Karsli and E. Ibikli, Rate of convergence of Chlodovsky type Bernstein operators for functions of bounded variation, Numer. Funct. Anal. Optim. 28(3–4) (2007), pp. 367–378; H. Karsli and E. Ibikli, Convergence rate of a new Bézier variant of Chlodovsky operators to bounded variation functions, J. Comput. Appl. Math. 212(2) (2008), pp. 431–443.] are essentially improved and extended to more general classes of functions.

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Acknowledgement

The authors are thankful to the referees for their valuable comments and suggestions leading to a better presentation of this article.

Additional information

Notes on contributors

Harun Karsli

Dedicated to Prof Paul Leo Butzer on the occasion of his 80th birthday.

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