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:$[mathcal\{L\}\_n(f)]\; (x)\; =\; sum\_\{k=0\}^infty\; \{(-1)^k\; frac\{x^k\}\{k!\}\; phi\_n^\{(k)\}(x)\; fleft(frac\{k\}\{n\}\; ight)\}$where $xin\; [0,b)subsetmathbb\{R\}$ ($b$ can be $infty$), $ninmathbb\{N\}$, and $(phi\_n)\_\{ninmathbb\{N$ is a sequence of functions defined on $[0,b]$ that have the following properties for all $n,kinmathbb\{N\}$:
#$phi\_ninmathcal\{C\}^infty\; [0,b]$. Alternatively, $phi\_n$ has a Taylor series on $[0,b)$.
#$phi\_n(0)\; =\; 1$
#$phi\_n$ is completely monotone, ie $(-1)^kphi\_n^\{(k)\}geq\; 0$.
#There is an integer $c$ such that $phi\_n^\{(k+1)\}\; =\; -nphi\_\{n+c\}^\{(k)\}$ whenever $n>max\{0,-c\}$They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.cite encyclopedia|last=Agrawal|first=P. N.|editor=Michiel Hazewinkel|year=2001|title=Baskakov operators|encyclopedia=Encyclopaedia of Mathematics|publisher=Springer|isbn=1402006098|url=http://eom.springer.de/b/b110150.htm]

Basic results

The Baskakov operators are linear and positive.cite encyclopedia|last=Agrawal|first=P. N.|coauthors=T. A. K. Sinha|editor=Michiel Hazewinkel|year=2001|title=Bernstein-Baskakov-Kantorovich operator|encyclopedia=Encyclopaedia of Mathematics|publisher=Springer|isbn=1402006098|url=http://eom.springer.de/b/b110350.htm]

References

*cite journal|last=Baskakov|first=V. A.|year=1957|title=An example of a sequence of linear positive operators in the space of continuous functions|journal=Proceedings of the USSR Academy of Sciences|volume=113|pages=249–251 ru icon

Footnotes