Favard operator

Short description: Functional analysis operator

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

[math]\displaystyle{ [\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)} }[/math]

where [math]\displaystyle{ x\in\mathbb{R} }[/math], [math]\displaystyle{ n\in\mathbb{N} }[/math]. They are named after Jean Favard.

Generalizations

A common generalization is:

[math]\displaystyle{ [\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)} }[/math]

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

References

• Favard, Jean (1944). "Sur les multiplicateurs d'interpolation" (in fr). Journal de Mathématiques Pures et Appliquées 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).[1]