TY - JOUR
AU - Stancu, Dimitrie D.
PY - 2001/02/01
Y2 - 2024/03/01
TI - On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu
JF - Rev. Anal. Numér. Théor. Approx.
JA - Rev. Anal. Numér. Théor. Approx.
VL - 30
IS - 1
SE - Articles
DO - 10.33993/jnaat301-687
UR - https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art13
SP - 95-105
AB - <p>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<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 \(\left( T_mf\right)\) 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 \(2sr\leq m\).</p><p>We give an integral expression for the remainders, as well as a representation by using divided differences of second order.</p>
ER -