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

where ( can be ), , and is a sequence of functions defined on that have the following properties for all :

  1. . Alternatively, has a Taylor series on .
  2. is completely monotone, i.e. .
  3. There is an integer such that whenever

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) [1994], "Baskakov operators", in Michiel Hazewinkel (ed.), Encyclopedia of Mathematics, EMS Press, ISBN 1-4020-0609-8
  2. 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
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