Baskakov operator

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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 $n,k\in\mathbb{N}$ :

$\phi_n\in\mathcal{C}^\infty[0,b]$ . Alternatively, $φ n$ has a Taylor series on $[0, b)$ .
$φ n (0) = 1$
$φ n$ is completely monotone, ie $(-1)^k\phi_n^{(k)}\geq 0$ .
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]

[edit] Basic results

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

[edit] References

Baskakov, V. A. (1957). "An example of a sequence of linear positive operators in the space of continuous functions". Proceedings of the USSR Academy of Sciences 113: 249-251. (Russian)

[edit] Footnotes

^ Agrawal, P. N. (2001). "Baskakov operators". Encyclopaedia of Mathematics. Ed. Michiel Hazewinkel. Springer. ISBN 1402006098.
^ Agrawal, P. N.; T. A. K. Sinha (2001). "Bernstein-Baskakov-Kantorovich operator". Encyclopaedia of Mathematics. Ed. Michiel Hazewinkel. Springer. ISBN 1402006098.