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

Abstract

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