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.^[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.