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 ${\displaystyle x\in [0,b)\subset \mathbb {R} }$ ($b$ can be $\infty$), $n\in \mathbb {N}$, 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, $\phi _{n}$ has a Taylor series on ${\displaystyle [0,b)}$.
2. ${\displaystyle \phi _{n}(0)=1}$
3. $\phi _{n}$ is completely monotone, i.e. ${\displaystyle (-1)^{k}\phi _{n}^{(k)}\geq 0}$.
4. There is an integer $c$ 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]

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.

Footnotes

1. Agrawal, P. N. (2001). "Baskakov operators". In Michiel Hazewinkel. Encyclopaedia of Mathematics. Springer. ISBN 1-4020-0609-8.
2. Agrawal, P. N.; T. A. K. Sinha (2001). "Bernstein–Baskakov–Kantorovich operator". In Michiel Hazewinkel. Encyclopaedia of Mathematics. Springer. ISBN 1-4020-0609-8.