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

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.