# On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu

## Abstract

In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation. By using the theory of polynomials of binomial type \((p_m)\) he has associated to a function \(f\in C[0,1]\) a linear operator defined by the formula

\[

\left( T_m f\right) (x) = \tfrac{1}{p_m(1)} \textstyle\sum\limits _{k=0} ^m \tbinom{m}{k}

p_k (x) p_{m-k} (1-x) f\big(\tfrac{k}{m}\big).

\]

Examples of such operators were considered in several subsequent papers.

In this paper we present a convergence theorem corresponding to the sequence \(\left( T_mf\right)\) and we also present a more general sequence of operators of approximation \(S_{m,r,s}\), where \(r\) and \(s\) are nonnegative integers such that \(2sr\leq m\).

We give an integral expression for the remainders, as well as a representation by using divided differences of second order.

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## Published

## How to Cite

*Rev. Anal. Numér. ThéOr. Approx.*,

*30*(1), 95-105. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art13