Voronovskaya-Type Theorems for Derivatives ofthe Bernstein-Chlodovsky Polynomials and theSzasz-Mirakyan Operator( Dedicated to the late Professor Wladyslaw Orlicz~1 (1903-1990) incommemoration of his doctoral thesis of 1928 as well as to his PoznanMathematical School )

摘要

This paper is devoted to a study of a Voronovskaya-type theorem forthe derivative of the Bernstein-Chlodovsky polynomials and to a comparison of itsapproximation effectiveness with the corresponding theorem for the much better-known Szasz-Mirakyan operator. Since the Chlodovsky polynomials contain a fac-tor b_n,tending to infinity having a certain degree of freedom, these polynomials turnout to be generally more efficient in approximating the derivative of the associatedfunction than does the Szasz operator. Moreover, whereas Chlodovsky polynomialsapply to functions which are even of order O(exp(x~P))for anyp >1, the Szasz-Mirakyan operator does so only for p = 1; it diverges for p > 1. The proofs employbut refine practical methods used by Jerzy Albrycht and Jerzy Radecki ( in paperswhich are almost never cited ) as well as by further mathematicians from the greatPoznan school.