Kolmogorov widths of weighted Sobolev classes on a domain for a special class of weights
Let \(X,Y\) be a normed spaces, \(M\subset X\). For \(n \in \mathbb{Z}\), let \(\mathcal {L}_n(X)\) be the family of all subspaces of \(X\) of dimension \(n\). The Kolmogorov \(n\)-width and the linear \(n\)-width are defined by \[ d_n(M,X)=\inf_{L\in \mathcal{L}_n(X)}\sup_{x\in M}\inf_{y\in L}\|x-y\|, \quad \lambda _n(M,X)=\inf_{A\in L(X,Y),\text{rk}A\leq n}\sup_{x\in M}\|x-Ax\|, \] respectively. For \(A\in L(X,Y)\), the approximation numbers are defined by \[ {\mathcal A} _n(A)=\inf_{A_n\in L(X,Y),\text{ rk} A_n\leq n}\sup_{x\in M}\|A-A_n\|. \] If \(X\) and \(Y\) represent certain weighted Sobolev classes and corresponding weighted Lebesgue spaces over a bounded domain \(\Omega \) in \(\mathbb{R}^d\), respectively, such that the embedding is compact, relations between Kolmogorov \(n\)-width, linear \(n\)-width and approximation numbers are obtained.
scientific article; zbMATH DE number 6097095
scientific article