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

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

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

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

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

Basic results

The Baskakov operators are linear and positive.

References

• 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.