Local approximation from spline spaces on box meshes
Piecewise polynomial approximations -- especially if the choice of the very useful quasi-interpolation method with splines or otherwise (e.g., kernel based methods) is made -- are well-studied ways to approximate functions in many variables. In this paper, the famous univariate piecewise polynomials (splines) are generalised to the multivariate situation by using tensor-product splines (other extensions are available). Error estimates (i.e., more precisely, approximation orders) are in the particular focus of the authors in this article when quasi-interpolation is used and in particular when so-called boxed meshes are employed. Remarkably, the authors are able to provide both \textit{local} and \textit{global} error estimates and those in the very general context of Sobolev norms. In particular, error bounds of approximations and their derivatives are derived. In some special cases, those error bounds are exponentially decreasing with the degree of the piecewise polynomials increasing, otherwise they are asymptotic with respect to point distances becoming smaller as is normal in the classical way. The results are quite general as error estimates are provided in the \(L_p/L_q\) fashion, i.e., approximation errors and their partial derivatives in \(L_p\) are estimated in \(L_q\) norms. Some of the estimates are isotropic (independent of direction), some depend on directions (anisotropic).
scientific article; zbMATH DE number 7371334
scientific article