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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 \( n,k\in\mathbb{N} \):
\( \phi_n\in\mathcal{C}^\infty[0,b] \). 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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