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In [[functional analysis]], a branch of [[mathematics]], the '''Favard operators''' are defined by:
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:<math>[\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)}</math>
 
where <math>x\in\mathbb{R}</math>, <math>n\in\mathbb{N}</math>, and <math>c\in\mathbb{R^{+}}</math>.<ref name="Nowak">{{cite journal| last=Nowak | first=Grzegorz | coauthors=Aneta Sikorska-Nowak | date=14 November 2007| title=On the generalized Favard–Kantorovich and Favard–Durrmeyer operators in exponential function spaces | journal=Journal of Inequalities and Applications | volume=2007 | url=http://www.hindawi.com/journals/jia/raa.75142.html | doi=10.1155/2007/75142 | pages=1| format={{dead link|date=June 2010}}}}</ref> They are named after [[Jean Favard]].
 
==Generalizations==
A common generalization is:
:<math>[\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>(\gamma_n)_{n=1}^\infty</math> is a positive sequence that [[Limit of a sequence|converges]] to 0.<ref name="Nowak"/> This reduces to the classical Favard operators when <math>\gamma_n^2=c/(2n)</math>.
 
==References==
*{{cite journal| last=Favard | first=Jean | authorlink=Jean Favard | year=1944 | title=Sur les multiplicateurs d'interpolation | journal=Journal de Mathematiques Pures et Appliquees | volume=23 | issue=9 | pages=219–247}} {{fr icon}} This paper also discussed [[Szász–Mirakyan operator]]s, which is why Favard is sometimes credited with their development (e.g. Favard–Szász operators).
 
===Footnotes===
<references/>
 
{{DEFAULTSORT:Favard Operator}}
[[Category:Approximation theory]]
 
 
{{mathanalysis-stub}}

Latest revision as of 00:59, 25 September 2014

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