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

Paul Leo Butzer, and Harun Karsli. "Voronovskaya-Type Theorems for Derivatives of the Bernstein-Chlodovsky Polynomials and the Szász-Mirakyan Operator." Commentationes Mathematicae 49.1 (2009): null. <http://eudml.org/doc/291814>.

@article{PaulLeoButzer2009,
abstract = {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 $p\ge 1$, the Szász–Mirakyan operator does so only for $p = 1$; it diverges for $p 1$. 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.},
author = {Paul Leo Butzer, Harun Karsli},
journal = {Commentationes Mathematicae},
keywords = {Bernstein–Chlodovsky polynomials; Szász–Mirakyan operator; Favard operator; Voronovskaya-type theorems},
language = {eng},
number = {1},
pages = {null},
title = {Voronovskaya-Type Theorems for Derivatives of the Bernstein-Chlodovsky Polynomials and the Szász-Mirakyan Operator},
url = {http://eudml.org/doc/291814},
volume = {49},
year = {2009},
}

TY - JOUR
AU - Paul Leo Butzer
AU - Harun Karsli
TI - Voronovskaya-Type Theorems for Derivatives of the Bernstein-Chlodovsky Polynomials and the Szász-Mirakyan Operator
JO - Commentationes Mathematicae
PY - 2009
VL - 49
IS - 1
SP - null
AB - 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 $p\ge 1$, the Szász–Mirakyan operator does so only for $p = 1$; it diverges for $p 1$. 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.
LA - eng
KW - Bernstein–Chlodovsky polynomials; Szász–Mirakyan operator; Favard operator; Voronovskaya-type theorems
UR - http://eudml.org/doc/291814
ER -