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

Abstract

In this article, we consider the Chlodovsky polynomials C _n f and their Bézier variants C _n,α 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,α 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.

Keywords:

• rate of convergence
• Bernstein–Chlodovsky polynomials
• Bézier basis
• Chanturiya's modulus of variation
• p-th power variation

AMS Subject Classifications::

• 41A25
• 41A35
• 41A36

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.