Favard operator explained

In functional analysis, a branch of mathematics, the Favard operators are defined by:

[ l{F} _{n(f)](x)} = (1)/(\sqrt{n\pi) } \sum_^\infty

where

x\in R , n\in N . They are named after Jean Favard.

Generalizations

A common generalization is:

[ l{F} _{n(f)](x)} = (1)/(n\gamma _{n\sqrt{2\pi}) } \sum_^\infty

where

(\gamma _{n)} ^{inf ty} ^{n =1} is a positive sequence that converges to 0.^[1] This reduces to the classical Favard operators when \gamma ^{2=1 /( 2 n)} _{n} .

References

• Favard . Jean . Jean Favard . 1944 . Sur les multiplicateurs d'interpolation . Journal de Mathématiques Pures et Appliquées . 23 . 9 . 219–247. fr. This paper also discussed Szász–Mirakyan operators, which is why Favard is sometimes credited with their development (e.g. Favard–Szász operators).https://gallica.bnf.fr/ark:/12148/bpt6k9704240z/f235.item

Footnotes

1. Nowak. Grzegorz. Aneta Sikorska-Nowak. 14 November 2007. On the generalized Favard–Kantorovich and Favard–Durrmeyer operators in exponential function spaces. Journal of Inequalities and Applications. 2007. 075142 . 10.1155/2007/75142. free.