Operators of discrete type
Let \(I_n\subset \mathbb{N}_0:=\{0\}\cup \mathbb{N}\) be a set of indices. Let \((L_n)_{n\ge 1}\) be a sequence of linear positive operators.
Considering a net \((x_{n,k})_{k\in I_n}\) on the interval \(I\subset \mathbb{R}\), a discrete operator is expressed by the relation
$$(L_n f)(x)=\sum_{k\in I_n}p_{n,k}(x)f(x_{n,k}),$$
where \(p_{n,k}\), \(k\in I_n\), are continuous positive functions on \(I\). Most commonly, equidistant nodes are chosen, for example
\(x_{n,k}=\displaystyle\frac{k}{n}\), \(k\in I_n\).
Here are the most famous classic examples.
1. The Bernstein polynomials
$$(B_n f)(x)=\sum_{k=0}^n { n \choose k}x^k(1-x)^{n-k}f\left(\frac{k}{n}\right),\ n\ge 1,\ f\in C([0,1]),$$
which has the property \(\lim\limits_{n\to \infty }B_n f=f\) uniformly on the interval \([0,1]\).
2. The Sz\’asz-Mirakjan operators
$$(S_n f)(x)=e^{-nx}\sum_{k=0}^\infty \frac{n^k x^k}{k!}f\left(\frac{k}{n}\right),\
x\ge 0,\ n\ge 1,$$
for \(f\in E_2:=\left\{f\in C([0,\infty )),\ \displaystyle\frac{f(x)}{1+x^2} \mbox{ is convergent as } x\to \infty \right\}\).
For any fix \(b>0\), we have \(\lim\limits_{n\to \infty }S_n f=f\) uniformly on \([0,b]\).