2025/07/01 by Yurii Kolomoitsev, Kolomoitsev, Yurii
Mathematics · #Approximation Theory and Sequence Spaces #Mathematical Approximation and Integration #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2507.00667
Traditional measures of smoothness often fail to provide accurate Lp-error estimates for approximation by sampling or interpolation operators, especially for functions with low smoothness. To address this issue, we introduce a modified measure of smoothness that incorporates the local behavior of a function at the sampling points through the use of averaged operators. With this new tool, we obtain matching direct and inverse error estimates for a wide class of sampling operators and functions in Lp spaces. Additionally, we derive a criterion for the convergence of sampling operators in Lp, identify conditions that ensure the exact rate of approximation, construct realizations of K-functionals based on these operators, and study the smoothness properties of sampling operators. We also demonstrate how our results apply to several well-known operators, including the classical Whittaker-Shannon sampling operator, sampling operators generated by B-splines, and those based on the Gaussian.