Favard operator

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

[ℱ_{n} (f)] (x) = \frac{1}{\sqrt{n π}} \sum_{k = - \infty}^{\infty} \exp (- n {(\frac{k}{n} - x)}^{2}) f (\frac{k}{n})

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

Generalizations

A common generalization is:

[ℱ_{n} (f)] (x) = \frac{1}{n γ_{n} \sqrt{2 π}} \sum_{k = - \infty}^{\infty} \exp (\frac{- 1}{2 γ_{n}^{2}} {(\frac{k}{n} - x)}^{2}) f (\frac{k}{n})

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

References

Favard, Jean (1944). "Sur les multiplicateurs d'interpolation". Journal de Mathématiques Pures et Appliquées (in French). 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).

Footnotes

↑ Nowak, Grzegorz; Aneta Sikorska-Nowak (14 November 2007). "On the generalized Favard–Kantorovich and Favard–Durrmeyer operators in exponential function spaces". Journal of Inequalities and Applications. 2007: 075142. doi:10.1155/2007/75142.

This mathematical analysis–related article is a stub. You can help Wikipedia by adding missing information.

[Nowak-1] Nowak, Grzegorz; Aneta Sikorska-Nowak (14 November 2007). "On the generalized Favard–Kantorovich and Favard–Durrmeyer operators in exponential function spaces". Journal of Inequalities and Applications. 2007: 075142. doi:10.1155/2007/75142.