Let \(I_n\subset \mathbb{N}_0:=\{0\}\cup \mathbb{N}\) be a set of indices. The most common cases are the following two: \(I_n=\{0,1,\ldots ,n\}\) and \(I_n=\mathbb{N}_0\).
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 \(I\), a continuous operator refers to the use of integrals in its definition.
The purpose of using these operators is to approximate functions from Lebesgue spaces \(L_p(I)\) or from their subspaces. There are two main constructions.
The first is of Kantorovich type
$$(L_n f)(x)=\sum_{k\in I_n}p_{n,k}(x)\int_{x_{n,k}}^{x_{n,k+1}}f(t)dt.$$
If \(x_{n,k+1}\not\in I_n\), then the integral is null. The second construction is of Durrmeyer type
$$(L_n f)(x)=\sum_{k\in I_n} p_{n,k}(x)\int _I p_{n,k}(t)f(t)dt.$$
In the above, usually \(p_{n,k}\in C(I)\), \(k\in I_n\), and they are positive functions enjoying the property \(\displaystyle\sum_{k\in I_n}p_{n,k}(x)=1\), \(x\in I\).