Favard operator

Favard operator

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

: [mathcal{F}_n(f)] (x) = frac{sqrt{n{nsqrt{cpi sum_{k=-infty}^infty {exp{left({frac{-n}{c} {left({frac{k}{n}-x} ight)}^2 } ight)} fleft(frac{k}{n} ight)}

where xinmathbb{R}, ninmathbb{N}, and cinmathbb{R^{+.cite journal| last=Nowak | first=Grzegorz | coauthors=Aneta Sikorska-Nowak | year=2007 | month=November | day=14 | 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] They are named after Jean Favard.

Generalizations

A common generalization is:: [mathcal{F}_n(f)] (x) = frac{1}{ngamma_nsqrt{2pi sum_{k=-infty}^infty {exp{left({frac{-1}{2gamma_n^2} {left({frac{k}{n}-x} ight)}^2 } ight)} fleft(frac{k}{n} ight)}

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

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 operators, which is why Favard is sometimes credited with their development (eg Favard-Szász operators).

Footnotes


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