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Voronovskaya-Type Theorems for Derivatives of the Bernstein-Chlodovsky Polynomials and the Szász-Mirakyan Operator

100%

Butzer P. L., Karsli H.

Commentationes Mathematicae

2009

tom 49

nr 1

This paper is devoted to a study of a Voronovskaya-type theorem for the derivative of the Bernstein–Chlodovsky polynomials and to a comparison of its approximation effectiveness with the corresponding theorem for the much better-known Szász–Mirakyan operator. Since the Chlodovsky polynomials contain a factor \(b_n\) tending to infinity having a certain degree of freedom, these polynomials turn out to be generally more efficient in approximating the derivative of the associated function than does the Szász operator. Moreover, whereas Chlodovsky polynomials apply to functions which are even of order \(O(\text{exp}(x^p))\) for any \(p\geq 1\), the Szász–Mirakyan operator does so only for \(p = 1\); it diverges for \(p \gt 1\). The proofs employ but refine practical methods used by Jerzy Albrycht and Jerzy Radecki (in papers which are almost never cited ) as well as by further mathematicians from the great Poznań school.

On the rates of convergence of Chlodovsky-Kantorovich operators and their Bézier variant

100%

Pych-Taberska P., Karsli H.

Commentationes Mathematicae

2009

tom 49

nr 2

In the present paper we consider the Bézier variant of Chlodovsky-Kantorovich operators \(K_{n−1,\alpha} f\) for functions \(f\) measurable and locally bounded on the interval \([0,\infty)\). By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of \(K_{n−1,\alpha} f (x)\) at those \(x \gt 0\) at which the one-sided limits \(f (x+)\), \(f(x-)\) exist.

Ograniczanie wyników

2 Commentationes Mathematicae

2 Karsli H.

1 Butzer P. L.

1 Pych-Taberska P.

2 2009

Wyniki wyszukiwania

Voronovskaya-Type Theorems for Derivatives of the Bernstein-Chlodovsky Polynomials and the Szász-Mirakyan Operator

On the rates of convergence of Chlodovsky-Kantorovich operators and their Bézier variant