Favard operator

Short description: Functional analysis operator

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

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

where $x\in \mathbb{ℝ}$, $n\in \mathbb{\mathbb{N}}$. They are named after Jean Favard.

Generalizations

A common generalization is:

$[{{ℱ}}_n(f)](x)=\frac{1}{n{\gamma }_n\sqrt{2\pi }}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\frac{-1}{2{\gamma }_n^2}{\left(\frac{k}{n}-x\right)}^2\right)f\left(\frac{k}{n}\right)$

where $({\gamma }_n{)}_{n=1}^{\mathrm{\infty }}$ is a positive sequence that converges to 0. This reduces to the classical Favard operators when ${\gamma }_n^2=1/(2n)$.

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]