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
- [math]\displaystyle{ [\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)} }[/math]
where [math]\displaystyle{ x\in[0,b)\subset\mathbb{R} }[/math] ([math]\displaystyle{ b }[/math] can be [math]\displaystyle{ \infty }[/math]), [math]\displaystyle{ n\in\mathbb{N} }[/math], and [math]\displaystyle{ (\phi_n)_{n\in\mathbb{N}} }[/math] is a sequence of functions defined on [math]\displaystyle{ [0,b] }[/math] that have the following properties for all [math]\displaystyle{ n,k\in\mathbb{N} }[/math]:
- [math]\displaystyle{ \phi_n\in\mathcal{C}^\infty[0,b] }[/math]. Alternatively, [math]\displaystyle{ \phi_n }[/math] has a Taylor series on [math]\displaystyle{ [0,b) }[/math].
- [math]\displaystyle{ \phi_n(0) = 1 }[/math]
- [math]\displaystyle{ \phi_n }[/math] is completely monotone, i.e. [math]\displaystyle{ (-1)^k\phi_n^{(k)}\geq 0 }[/math].
- There is an integer [math]\displaystyle{ c }[/math] such that [math]\displaystyle{ \phi_n^{(k+1)} = -n\phi_{n+c}^{(k)} }[/math] whenever [math]\displaystyle{ n\gt \max\{0,-c\} }[/math]
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). (in Russian)Doklady Akademii Nauk SSSR 113: 249–251.
Footnotes
- ↑ Hazewinkel, Michiel, ed. (2001), "Baskakov operators", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- ↑ Hazewinkel, Michiel, ed. (2001), "Bernstein–Baskakov–Kantorovich operator", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
Original source: https://en.wikipedia.org/wiki/Baskakov operator.
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