Baskakov operator

In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators. They are defined by

${\displaystyle [{\mathcal {L}}_{n}(f)](x)=\sum _{k=0}^{\infty }{(-1)^{k}{\frac {x^{k}}{k!}}\phi _{n}^{(k)}(x)f\left({\frac {k}{n}}\right)}}${\displaystyle [{\mathcal {L}}_{n}(f)](x)=\sum _{k=0}^{\infty }{(-1)^{k}{\frac {x^{k}}{k!}}\phi _{n}^{(k)}(x)f\left({\frac {k}{n}}\right)}}

where ${\displaystyle x\in [0,b)\subset \mathbb {R} }${\displaystyle x\in [0,b)\subset \mathbb {R} } (${\displaystyle b}${\displaystyle b} can be ${\displaystyle \infty }${\displaystyle \infty }), ${\displaystyle n\in \mathbb {N} }${\displaystyle n\in \mathbb {N} }, and ${\displaystyle (\phi _{n})_{n\in \mathbb {N} }}${\displaystyle (\phi _{n})_{n\in \mathbb {N} }} is a sequence of functions defined on ${\displaystyle [0,b]}${\displaystyle [0,b]} that have the following properties for all ${\displaystyle n,k\in \mathbb {N} }${\displaystyle n,k\in \mathbb {N} }:

1. ${\displaystyle \phi _{n}\in {\mathcal {C}}^{\infty }[0,b]}${\displaystyle \phi _{n}\in {\mathcal {C}}^{\infty }[0,b]}. Alternatively, ${\displaystyle \phi _{n}}${\displaystyle \phi _{n}} has a Taylor series on ${\displaystyle [0,b)}${\displaystyle [0,b)}.
2. ${\displaystyle \phi _{n}(0)=1}${\displaystyle \phi _{n}(0)=1}
3. ${\displaystyle \phi _{n}}${\displaystyle \phi _{n}} is completely monotone, i.e. ${\displaystyle (-1)^{k}\phi _{n}^{(k)}\geq 0}${\displaystyle (-1)^{k}\phi _{n}^{(k)}\geq 0}.
4. There is an integer ${\displaystyle c}${\displaystyle c} such that ${\displaystyle \phi _{n}^{(k+1)}=-n\phi _{n+c}^{(k)}}${\displaystyle \phi _{n}^{(k+1)}=-n\phi _{n+c}^{(k)}} whenever ${\displaystyle n>\max\{0,-c\}}${\displaystyle n>\max\{0,-c\}}

They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.^[1]

The Baskakov operators are linear and positive.^[2]

• Baskakov, V. A. (1957). Пример последовательности линейных положительных операторов в пространстве непрерывных функций [An example of a sequence of linear positive operators in the space of continuous functions]. Doklady Akademii Nauk SSSR (in Russian). 113: 249–251.

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