A Brief History of the Favard Operator and Its Variants

Ulrich Abel ()
Additional contact information
Ulrich Abel: Fachbereich MND

A chapter in Mathematical Analysis, Approximation Theory and Their Applications, 2016, pp 1-13 from Springer

Abstract: 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 , σ n $$F_{n,\sigma _{n}}$$ of the Favard operator and its Durrmeyer variant F ̃ n , σ n $$\tilde{F }_{n,\sigma _{n}}$$ . 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 F n , σ n f x $$\left (F_{n,\sigma _{n}}f\right )\left (x\right )$$ and F ̃ n , σ n f x $$\left (\tilde{F } _{n,\sigma _{n}}f\right )\left (x\right )$$ 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.

Keywords: Approximation by positive operators; Rate of convergence; Degree of approximation. (search for similar items in EconPapers)
Date: 2016
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-31281-1_1

Ordering information: This item can be ordered from
http://www.springer.com/9783319312811

DOI: 10.1007/978-3-319-31281-1_1

Access Statistics for this chapter

More chapters in Springer Optimization and Its Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().