# Sheffer sequences, probability distributions and approximation operators

## Abstract

We present a new method to compute formulas for the action on monomials of a generalization of binomial approximation operators of Popoviciu type, or equivalently moments of associated discrete probability distributions with ﬁnite support.

These quantities are necessary to check the assumptions of the Korovkin Theorem for approximation operators, or equivalently the Feller Theorem for convergence of the probability distributions.

Our method uniﬁes and simpliﬁes computations of well-known special cases. It only requires a few basic facts from Umbral Calculus.

We illustrate our method to well-known approximation operators and probability distributions, as well as to some recent q-generalizations of the Bernstein approximation operator introduced by Lewanowicz and Wozny, Lupas and Phillips.

## Authors

M. Crăciun
(Tiberiu Popoviciu Institute of Numerical Analysis)
A. Di Bucchianico
(Department of Mathematics and Computer Science, Technische Universiteit Eindhoven)

## Keywords

approximation operators of Popoviciu type; moments; Umbral Calculus; Sheﬀer sequences; basic sequences; delta operators

### References

See the expanding block below.

## Paper coordinates

SPOR Report 2005-04, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven

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