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 $c\in\mathbb{R^{+}}$.^[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

• Favard, Jean (1944). "Sur les multiplicateurs d'interpolation". Journal de Mathematiques Pures et Appliquees (in French) 23 (9): 219–247. This paper also discussed Szász–Mirakyan operators, which is why Favard is sometimes credited with their development (e.g. Favard–Szász operators).

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: 1. doi:10.1155/2007/75142.