Boussinesq approximation (water waves): Difference between revisions

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In [[functional analysis]], a branch of [[mathematics]], the '''Baskakov operators''' are generalizations of [[Bernstein polynomials]], [[Szász–Mirakyan operator]]s, and [[Lupas operator]]s. They are defined by
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:<math>[\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>x\in[0,b)\subset\mathbb{R}</math> (<math>b</math> can be <math>\infty</math>), <math>n\in\mathbb{N}</math>, and <math>(\phi_n)_{n\in\mathbb{N}}</math> is a sequence of functions defined on <math>[0,b]</math> that have the following properties for all <math>n,k\in\mathbb{N}</math>:
#<math>\phi_n\in\mathcal{C}^\infty[0,b]</math>. Alternatively, <math>\phi_n</math> has a [[Taylor series]] on <math>[0,b)</math>.
#<math>\phi_n(0) = 1</math>
#<math>\phi_n</math> is completely monotone, i.e. <math>(-1)^k\phi_n^{(k)}\geq 0</math>.
#There is an integer <math>c</math> such that <math>\phi_n^{(k+1)} = -n\phi_{n+c}^{(k)}</math> whenever <math>n>\max\{0,-c\}</math>
They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.<ref name="Agrawal">{{cite encyclopedia|last=Agrawal|first=P. N.|editor=[[Michiel Hazewinkel]]|year=2001|title=Baskakov operators|encyclopedia=Encyclopaedia of Mathematics|publisher=Springer|isbn=1-4020-0609-8|url=http://eom.springer.de/b/b110150.htm}}</ref>


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The Baskakov operators are linear and positive.<ref name="Agrawal2">{{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=1-4020-0609-8|url=http://eom.springer.de/b/b110350.htm}}</ref>
 
==References==
*{{cite journal| last=Baskakov | first=V. A. | year=1957 | title = Пример последовательности линейных положительных операторов в пространстве непрерывных функций | trans_title=An example of a sequence of linear positive operators in the space of continuous functions | journal=[[Doklady Akademii Nauk SSSR]] | language=Russian | volume=113 | pages=249–251}}
 
===Footnotes===
<references/>
 
{{mathanalysis-stub}}
[[Category:Approximation theory]]

Revision as of 21:06, 28 February 2014

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