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