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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, $$\phi_n$$ has a Taylor series on [0,b).
$$\phi_n(0) = 1$$
$$\phi_n$$ is completely monotone, i.e. $$(-1)^k\phi_n^{(k)}\geq 0.$$
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]

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

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

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