Baskakov Operators

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_n(f)x) = \sum_^\infty
where x\in ,b)\subset\mathbb ($b$ can be $\infty$), $n\in\mathbb$, and $(\phi_n)_$ is a sequence of functions defined on [0,b/math> that have the following properties for all $n,k\in\mathbb$:
#$\phi_n\in\mathcal^\infty[0,b]$. Alternatively, $\phi_n$ has a Taylor series on $[0,b)$.
#$\phi_n(0) = 1$
#$\phi_n$ is completely monotone, i.e. $(-1)^k\phi_n^\geq 0$.
#There is an integer $c$ such that $\phi_n^ = -n\phi_^$ whenever $n>\max\$
They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.

Basic results

The Baskakov operators are linear and positive.

References

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Footnotes

{{mathanalysis-stub
Approximation theory