A Jackson Type Estimate for Shepard Operators in L ^p Spaces for p≧ 1

Abstract

This paper considers the approximation of the Kantorovich–Shepard operators in $L^p$ spaces for $p \geqq 1$. For $f\left( x \right) \in L_{\left[ {0,1} \right]}^p$ the Kantorovich–Shepard operators $L_{n,\lambda } \left( {f,x} \right)$ are defined by (1.1). Then

$$\left\| {L_{n,\lambda } \left( f \right) - f} \right\|_{L_{\left[ {0,1} \right]}^p } \leqq C_{p,\lambda } \omega \left( {f,\varepsilon _n } \right)_{L_{\left[ {0,1} \right]}^p } ,$$

where $C_{p,\lambda }$ is a positive number depending only on $p$ and λ, and $$ \varepsilon_{n} =\cases{ n^{-1}, & if \ $\lambda > 2$; \cr\nosm n^{-1}\log n, & if \ $\lambda = 2$; \cr\nosm n^{1-\lambda}, & if \ $1 < \lambda < 2$. \cr} $$