Quantitative Korovkin theorems for positive linear operators on $L_{p}$-spaces

Abstract:

Let $({L_n})$ be a sequence of positive linear operators on ${L_p}(\Omega )$, $1 \leqslant p < \infty$ or $C(\Omega )$ with $\Omega \subseteq {R^m}$. For suitable $\Omega$, the functions $({\varphi _i})_{i = 0}^{m + 1}$ given by ${\varphi _0}(x) \equiv 1$, ${\varphi _i}(x) \equiv {x_i}$, $1 \leqslant i \leqslant m$,and ${\varphi _{m + 1}}(x) \equiv {\left | x \right |^2}$ form a test set for ${L_p}(\Omega )$. That is, if ${L_n}({\varphi _i})$ converges to ${\varphi _i}$ in ${\left \| \cdot \right \|_p}$ for each $i = 0,1, \ldots ,m + 1$, then ${L_n}(f)$ converges to f in ${\left \| \cdot \right \|_p}$ for each $f \in {L_p}(\Omega )$. We give here quantitative versions of this result. Namely, we estimate ${\left \| {f - {L_n}f} \right \|_p}$ in terms of the error ${\left \| {{\varphi _i} - {L_n}{\varphi _i}} \right \|_p}$, $0 \leqslant i \leqslant m + 1$,and the smoothness of the function f.