Approximation operators constructed by means of Sheffer sequences

Abstract

In this paper we introduce a class of positive linear operators by using the "umbral calculus", and we study some approximation properties of it. Let \( Q \) be a delta operator, and \(S\) an invertible shift invariant operator. For \(f\in C[0,1]\) we define \begin{equation*} (L_{n}^{Q,S}f)(x)= {\textstyle\frac{1}{s_{n}(1)}} \sum\limits_{k=0}^{n}{\textstyle\binom{n}{k}}p_{k}(x)s_{n-k}(1-x)f\left( \tfrac{k}{n} \right), \end{equation*} where \((p_{n})_{n\geq0}\) is a binomial sequence which is the basic sequence for \(Q,\) and \((s_{n})_{n\geq0}\) is a Sheffer set, \( s_{n}=S^{-1}p_{n} \). These operators generalize the binomial operators of T. Popoviciu.