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

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

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

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

They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.{{SpringerEOM|last=Agrawal|first=P. N.|editor=Michiel Hazewinkel|year=2001|title=Baskakov operators|isbn=1-4020-0609-8}}