Gibbs derivatives

Abstract

In this paper we present a short account of the background to the theory of Gibbs differentiation on the occasion of the 25th anniversary of its introduction and the 20th and 15th anniversaries of the Butzer-Wagner dyadic derivative and Onneweer's extension of the theory to Vilenkin groups, respectively, each the most important result in one of two different ways of generalizing Gibbs’ introductory result.

We have also attempted to give a rather general characterization of Gibbs differentiators, regarded as a class of differential operators on various, not necessarily Abelian, groups.