Approximation operators constructed by means of Sheffer sequences

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 \((L_{n}^{Q,S}f)(x)=\frac{1}{sn_{(1)}}\sum \limits_{k=0}^{n}\binomial{n}{k} p_{k}(x)s_{n-k}(1-x)f(\frac{k}{n})\), 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.

Maria Craciun
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

approximation operators; Sheffer sequences; basic sequences; delta operators

https://ictp.acad.ro/craciun/papers/craciun-2001-ANTA.pdf

M. Crăciun, Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numér. Théor. Approx., vol. 30 (2001), no. 2, 135-150

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