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
[ L n ( f ) ] ( x ) = ∑ k = 0 ∞ ( − 1 ) k x k k ! ϕ n ( k ) ( x ) f ( k n ) {\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)}}
where x ∈ [ 0 , b ) ⊂ R {\displaystyle x\in [0,b)\subset \mathbb {R} } ( b {\displaystyle b} can be ∞ {\displaystyle \infty } ), n ∈ N {\displaystyle n\in \mathbb {N} } , and ( ϕ n ) n ∈ N {\displaystyle (\phi _{n})_{n\in \mathbb {N} }} is a sequence of functions defined on [ 0 , b ] {\displaystyle [0,b]} that have the following properties for all n , k ∈ N {\displaystyle n,k\in \mathbb {N} } :
ϕ n ∈ C ∞ [ 0 , b ] {\displaystyle \phi _{n}\in {\mathcal {C}}^{\infty }[0,b]} . Alternatively, ϕ n {\displaystyle \phi _{n}} has a Taylor series on [ 0 , b ) {\displaystyle [0,b)} .
ϕ n ( 0 ) = 1 {\displaystyle \phi _{n}(0)=1}
ϕ n {\displaystyle \phi _{n}} is completely monotone, i.e. ( − 1 ) k ϕ n ( k ) ≥ 0 {\displaystyle (-1)^{k}\phi _{n}^{(k)}\geq 0} . There is an integer c {\displaystyle c} such that ϕ n ( k + 1 ) = − n ϕ n + c ( k ) {\displaystyle \phi _{n}^{(k+1)}=-n\phi _{n+c}^{(k)}} whenever n > max { 0 , − c } {\displaystyle n>\max\{0,-c\}}
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 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.
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