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
 * $$[\mathcal{L}_n(f)](x) = \sum_{k=0}^\infty {(-1)^k \frac{x^k}{k!} \phi_n^{(k)}(x) f\left(\frac{k}{n}\right)}$$

where $$x\in[0,b)\subset\mathbb{R}$$ ($$b$$ can be $$\infty$$), $$n\in\mathbb{N}$$, and $$(\phi_n)_{n\in\mathbb{N}}$$ is a sequence of functions defined on $$[0,b]$$ that have the following properties for all $$n,k\in\mathbb{N}$$: They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.
 * 1) $$\phi_n\in\mathcal{C}^\infty[0,b]$$. Alternatively, $$\phi_n$$ has a Taylor series on $$[0,b)$$.
 * 2) $$\phi_n(0) = 1$$
 * 3) $$\phi_n$$ is completely monotone, i.e. $$(-1)^k\phi_n^{(k)}\geq 0$$.
 * 4) There is an integer $$c$$ such that $$\phi_n^{(k+1)} = -n\phi_{n+c}^{(k)}$$ whenever $$n>\max\{0,-c\}$$

Basic results
The Baskakov operators are linear and positive.