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
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} $$
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Xiao, W., Zhou, S. A Jackson Type Estimate for Shepard Operators in L p Spaces for p≧ 1. Acta Mathematica Hungarica 95, 217–224 (2002). https://doi.org/10.1023/A:1015684721815
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DOI: https://doi.org/10.1023/A:1015684721815