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]:

1. [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].
2. [math]\displaystyle{ \phi_n(0) = 1 }[/math]
3. [math]\displaystyle{ \phi_n }[/math] is completely monotone, i.e. [math]\displaystyle{ (-1)^k\phi_n^{(k)}\geq 0 }[/math].
4. 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.

Basic results

The Baskakov operators are linear and positive.

References

• Baskakov, V. A. (1957). (in Russian)Doklady Akademii Nauk SSSR 113: 249–251.