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On approximation by binomial operators of Tiberiu Popoviciu type. (English) Zbl 1007.41016
From the introduction: It is known that an important class of polynomial sequences $$(p_m)_{m\geq 0}$$ occurring in combinatorics and analysis is represented by the sequences of binomial type (B.T.), for which we have $$p_0$$, $$\text{deg} p_m=m$$ and the following equalities $p_m(u+v)=\sum_{k=0}^m{m \choose k}p_k(u) p_{m-k}(v) \tag{1.1}$ are identically satisfied in $$u$$ and $$v$$, for any nonnegative integer $$m$$.
T. Popoviciu introduced in [Bul. Soc. Stiint. Cluj 6, 146-148 (1931; Zbl 0002.39801)] an operator $$T_m$$ associated to a function $$f\colon [0,1]\to {\mathbb R}$$, by means of the formula $(T_mf)(x)=\frac{1}{p_m(1)} \sum \limits_{k=0}^m{m \choose k} p_k(x) p_{m-k}(1-x) f\left(\frac{k}{m}\right), \tag{1.5}$ where $$x\in [0,1]$$ and $$m\in {\mathbb N}$$, with the assumption: $$p_m(1)\neq 0$$.
In this paper we consider some more general binomial-type operators and we study their approximation properties, including the estimation of orders of approximations by means of the first and second moduli of smoothness, as well as representation of the remainder term in approximation formulas by the Tiberiu Popoviciu-type operators.

MSC:
 41A35 Approximation by operators (in particular, by integral operators) 41A63 Multidimensional problems
Zbl 0002.39801