Abstract
Given a function f on a bounded open subset Ω of \({\mathbb{R}}^n\) with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of \(\overline\Omega\) . This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.
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Arcangéli, R., López de Silanes, M.C. & Torrens, J.J. An extension of a bound for functions in Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing. Numer. Math. 107, 181–211 (2007). https://doi.org/10.1007/s00211-007-0092-z
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DOI: https://doi.org/10.1007/s00211-007-0092-z