Let \(U,V\) be vectorial spaces over a field \(K\). A function \(L:U\to V\) is linear operator if
$$L(x+y)=L(x)+L(y),\ \forall \ x,y\in V\quad \mbox{(additivity)},$$
$$L(\alpha x)=\alpha L(x),\ \forall \alpha \in K,\ \forall x\in V
\quad \mbox{(omogeneity)}.$$
If \(U\) is a space of real valued functions, \(L:U\to V\) is positive operator if
$$Lf\ge 0,\ \forall f\ge 0,\ f\in U.$$
More generally, let \(E\) and \(F\) be two function spaces defined on the same locally compact Hausdorff space such that \(E\subseteq F\).
A positive linear approximation process on \(E\) with respect to \(F\) is a net \((L_n)_{n\ge 1}\) of positive linear operators from \(E\) into \(F\) such that for every \(f\) belonging to \(E\), \((L_n f)_{n\ge 1}\)
converges to \(f\), the convergence being understood with respect to a suitable topology on \(F\).
In one-dimensional case, these sequences usually act in the spaces \(C(I)\), \(C_B(I)\), \(L_p(I)\) or in weighted spaces.
In the above \(I\subseteq \mathbb{R}\) is a compact interval or an unbounded interval, for example \(I=[0,\infty )\), \(I=\mathbb{R}\).