Favard operator

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

[\mathcal{F}_n(f)](x) = \frac{\sqrt{n}}{n\sqrt{c\pi}} \sum_{k=-\infty}^\infty {\exp{\left({\frac{-n}{c} {\left({\frac{k}{n}-x}\right)}^2 }\right)} f\left(\frac{k}{n}\right)}

where x\in\mathbb{R}, n\in\mathbb{N}, and c\in\mathbb{R^{%2B}}.[1] They are named after Jean Favard.

Generalizations

A common generalization is:

[\mathcal{F}_n(f)](x) = \frac{1}{n\gamma_n\sqrt{2\pi}} \sum_{k=-\infty}^\infty {\exp{\left({\frac{-1}{2\gamma_n^2} {\left({\frac{k}{n}-x}\right)}^2 }\right)} f\left(\frac{k}{n}\right)}

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

References

Footnotes

  1. ^ a b 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: 1. doi:10.1155/2007/75142. http://www.hindawi.com/journals/jia/raa.75142.html.