Special measures of smoothness for approximation by sampling operators in $L_p(\Bbb{R}^d)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915367819935744 |
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| author | Kolomoitsev, Yurii |
| author_facet | Kolomoitsev, Yurii |
| contents | Traditional measures of smoothness often fail to provide accurate $L_p$-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 $L_p$ spaces. Additionally, we derive a criterion for the convergence of sampling operators in $L_p$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Special measures of smoothness for approximation by sampling operators in $L_p(\Bbb{R}^d)$ Kolomoitsev, Yurii Numerical Analysis Classical Analysis and ODEs 41A05, 41A15, 41A17, 41A25, 41A27 Traditional measures of smoothness often fail to provide accurate $L_p$-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 $L_p$ spaces. Additionally, we derive a criterion for the convergence of sampling operators in $L_p$, 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. |
| title | Special measures of smoothness for approximation by sampling operators in $L_p(\Bbb{R}^d)$ |
| topic | Numerical Analysis Classical Analysis and ODEs 41A05, 41A15, 41A17, 41A25, 41A27 |
| url | https://arxiv.org/abs/2507.00667 |