Approximation operators constructed by means of Sheffer sequences. (English) Zbl 1027.41023

Crăciun, Maria

Rev. Anal. Numér. Théor. Approx. 30, No. 2, 135-150 (2001).

Summary: In this paper we introduce a class of positive linear operators by using the umbral calculus, and we study some approximation properties of them. Let \(Q\) be a delta operator, and \(S\) an invertible shift invariant operator. For \(f\in C[0,1]\) we define \[ (L_n^{Q,S}f)(x)=\frac{1}{s_n(1)} \sum_{k=0}^n{n\choose k}p_k(x)s_{n-k}(1-x)f(k/n), \] where \((p_n)_{n\geq 0}\) is a binomial sequence that is the basic sequence for \(Q\), and \((s_n)_{n\geq 0}\) is a Sheffer set, \(s_n=S^{-1}p_n\). These operators generalize the binomial operators of T. Popoviciu.

MSC:

41A36	Approximation by positive operators
05A40	Umbral calculus