Favard operator - Alchetron, The Free Social Encyclopedia

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

[ F n ( f ) ] ( x ) = n n c π ∑ k = − ∞ ∞ exp ⁡ ( − n c ( k n − x ) 2 ) f ( k n )

where x ∈ R , n ∈ N , and c ∈ R + . 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 )

where ( γ n ) n = 1 ∞ is a positive sequence that converges to 0. This reduces to the classical Favard operators when γ n 2 = c / ( 2 n ) .

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