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

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

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

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

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

Basic results

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