- Source: Favard operator
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).[1]
= Footnotes
=Kata Kunci Pencarian:
- Favard operator
- Szász–Mirakyan operator
- Favard's theorem
- Jean Favard
- List of numerical analysis topics
- Mark Krein
- Landau–Kolmogorov inequality
- List of polynomial topics
- Naum Akhiezer
- Sobolev orthogonal polynomials
