In functional analysis, a branch of mathematics, the Favard operators are defined by:
[ F n ( f ) ] ( x ) = 1 n π ∑ k = − ∞ ∞ exp ( − n ( k n − x ) 2 ) f ( k n ) {\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 x ∈ R {\displaystyle x\in \mathbb {R} } , n ∈ N {\displaystyle n\in \mathbb {N} } . They are named after Jean Favard.
Generalizations A common generalization is:
[ F n ( f ) ] ( x ) = 1 n γ n 2 π ∑ k = − ∞ ∞ exp ( − 1 2 γ n 2 ( k n − x ) 2 ) f ( k n ) {\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 ( γ n ) n = 1 ∞ {\displaystyle (\gamma _{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 ) {\displaystyle \gamma _{n}^{2}=1/(2n)} .
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).
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