@article{Stancu_2001, title={On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu}, volume={30}, url={https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art13}, DOI={10.33993/jnaat301-687}, abstractNote={<p>In 1931, Tiberiu Popoviciu has initiated a procedure for the construction&nbsp;of sequences of linear positive operators of approximation. By using the&nbsp;theory of polynomials of binomial type \((p_m)\) he has associated to a&nbsp;function \(f\in C[0,1]\) a linear operator defined by the formula<br>\[<br>\left( T_m f\right) (x) = \tfrac{1}{p_m(1)} \textstyle\sum\limits _{k=0} ^m \tbinom{m}{k}<br>p_k (x) p_{m-k} (1-x) f\big(\tfrac{k}{m}\big).<br>\]<br>Examples of such operators were considered in several subsequent papers.</p> <p>In this paper we present a convergence theorem corresponding to the sequence&nbsp;\(\left( T_mf\right)\) and we also present a more general sequence of operators&nbsp;of approximation \(S_{m,r,s}\), where \(r\) and \(s\) are nonnegative integers&nbsp;such that \(2sr\leq m\).</p> <p>We give an integral expression for the remainders, as well as a representation&nbsp;by using divided differences of second order.</p>}, number={1}, journal={Rev. Anal. Numér. Théor. Approx.}, author={Stancu, Dimitrie D.}, year={2001}, month={Feb.}, pages={95–105} }