Favard operator

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

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

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

Generalizations

A common generalization is:

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

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