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

• Autorzy: Paul Leo Butzer , Harun Karsli
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Bernstein–Chlodovsky polynomials; Szász–Mirakyan operator; Favard operator; Voronovskaya-type theorems

• Wydawca: Polish Mathematical Society
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2009
• Tom: 49
• Numer: 1

Daty

wydano

2009

online

2017-12-19

Twórcy

autor

Paul Leo Butzer

autor

Harun Karsli

Identyfikatory

DOI

10.14708/cm.v49i1.5277