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
=\sum _{k=0}^{\infty }{(-1)^{k}{\frac {x^{k}}{k!}}\phi _{n}^{(k)}(x)f\left({\frac {k}{n}}\right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d562980a1646ed0f71d7040ca21709ad7d3a99)
where ( can be ), , and is a sequence of functions defined on that have the following properties for all :
![{\displaystyle [0,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22e25e8f8604f012c599a7d4962562c4bb3f02)
- . Alternatively, has a Taylor series on .
- is completely monotone, i.e. .
- There is an integer such that whenever
![{\displaystyle \phi _{n}\in {\mathcal {C}}^{\infty }[0,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c965deec162dd6f9f7d25cda60c846ae816bd1)
They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.[1]
Basic results[edit]
The Baskakov operators are linear and positive.[2]
References[edit]
- 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[edit]
- ^ Agrawal, P. N. (2001) [1994], "Baskakov operators", in Michiel Hazewinkel (ed.), Encyclopedia of Mathematics, EMS Press, ISBN 1-4020-0609-8
- ^ Agrawal, P. N.; T. A. K. Sinha (2001) [1994], "Bernstein–Baskakov–Kantorovich operator", in Michiel Hazewinkel (ed.), Encyclopedia of Mathematics, EMS Press, ISBN 1-4020-0609-8