A Brief History of the Favard Operator and Its Variants

Abstract

In the year 1944 the well-known French mathematician Jean Favard (1902–1965) introduced a discretely defined operator which is a discrete analogue of the familiar Gauss–Weierstrass singular convolution integral. In the present chapter we sketch the history of this approximation operator during the past 70 years by presenting known results on the operator and its various extensions and variants. The first part after the introduction is dedicated to saturation of the classical Favard operator in weighted Banach spaces. Furthermore, we discuss the asymptotic behaviour of a slight generalization $F_{n,\sigma _{n}}$ of the Favard operator and its Durrmeyer variant $\tilde{F }_{n,\sigma _{n}}$. In particular, the local rate of convergence when applied to locally smooth functions is considered. The main result of this part consists of the complete asymptotic expansions for the sequences $\left (F_{n,\sigma _{n}}f\right )\left (x\right )$ and $\left (\tilde{F } _{n,\sigma _{n}}f\right )\left (x\right )$ as n tends to infinity. Furthermore, these asymptotic expansions are valid also with respect to simultaneous approximation. A further part is devoted to the recent work of several Polish mathematicians on approximation in weighted function spaces. Finally, we define left quasi-interpolants for the Favard operator and its Durrmeyer variant in the sense of Sablonnière.