\(L_ 2(\mathbb{R}^ n)\)-extremal problems on the basis of incomplete information
The paper under review deals with the best approximation problem for a densely defined, Banach space valued linear operator. Its purpose is to extend solutions of the best approximation problem from Sobolev spaces on the torus and the real line to the multivariate case by means of polyharmonic splines. The \(m\)th-polyharmonic splines on the space \(\mathbb{R}^n\) are annihilated by the \(m\)-th iterated Laplacian \(\Delta^m\) on the points of \(\mathbb{R}^n\), with exception of those points which belong to the lattice \(\mathbb{Z}^n\). For even numbers \(m\), they give rise to the cardinal interpolation problem on the space \(\mathbb{R}^n\). -- The paper is written in the technical vein characteristic for multivariate approximation, and the theory of function spaces on \(\mathbb{R}^n\). A constituent ingredient is formed by the Hardy-Littlewood-Pólya inequality which follows from the Plancherel formula in conjunction with Hölder's inequality.
scientific article; zbMATH DE number 808263
scientific article