Optimal spline spaces of higher degree for \(L^2\) \(n\)-widths
In this article, Kolmogorov \(n\)-width are studied, that is, the \(n\)-width of a subset \(A\), say, of a normed linear space \((X,\|\cdot\|)\) with respect to that space \(X\). The \(n\) comes from the dimension \(n\) of subspaces \(X_n\) of \(X\) which are used to define the \(n\)-width by the expression \[ \inf_{X_n}\sup_{u\in A}\inf_{v\in X_n}\|u-v\|. \] Now, subspaces \(X_n\) are called ``optimal'' if the \(n\)-width is attained by using them in place of the \(\inf_{X_n}\). The authors show in this paper that spline subspaces of high degrees provide optimal subspaces for \(A\)s given by Sobolev spaces (with or without zero boundary conditions) and \(X\) being spaces of square-integrable functions on the unit interval.
scientific article; zbMATH DE number 6690408
scientific article