Favard operator

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


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

where $$x\in\mathbb{R}$$, $$n\in\mathbb{N}$$. 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. This reduces to the classical Favard operators when $$\gamma_n^2=1/(2n)$$.