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

$[{{ℒ}}_n(f)](x)=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k\frac{x^k}{k!}{\varphi }_n^{(k)}(x)f\left(\frac{k}{n}\right)$

where $x\in [0,b)\subset \mathbb{ℝ}$ ($b$ can be $\mathrm{\infty }$), $n\in \mathbb{\mathbb{N}}$, and $({\varphi }_n{)}_{n\in \mathbb{\mathbb{N}}}$ is a sequence of functions defined on $[0,b]$ that have the following properties for all $n,k\in \mathbb{\mathbb{N}}$:

1. ${\varphi }_n\in {{𝒞}}^{\mathrm{\infty }}[0,b]$. Alternatively, ${\varphi }_n$ has a Taylor series on $[0,b)$.
2. ${\varphi }_n(0)=1$
3. ${\varphi }_n$ is completely monotone, i.e. $(-1{)}^k{\varphi }_n^{(k)}\ge 0$.
4. There is an integer $c$ such that ${\varphi }_n^{(k+1)}=-n{\varphi }_{n+c}^{(k)}$ whenever $n>\max \{0,-c\}$

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.{{cite journal}}: CS1 maint: unrecognized language (link)