Kolmogorov widths in finite-dimensional spaces with mixed norms
For \(1\leq p\), \(q\leq\infty\), \(n,m\in\mathbb{N}\), we introduce the following norm in the space \(\mathbb{R}^{mn}\): \[ |x|_{\ell^{n,m}_{p,q}}=\begin{cases} (\sum^m_{s=1}(\sum_{k\in\Delta_s}|x_k|^p)^{q/p})^{1/q}, & \text{if }1\leq p, q<\infty,\\ \max_{1\leq s\leq m}(\sum_{k\in \Delta_s}|x_k|^p)^{1/p}, &\text{if }q=\infty,\end{cases} \] where \(\Delta_s= \{k\in\mathbb{N}:(s- 1)n<k\leq sn\}\), \(s=1,\dots,m\). Thus, we get a normed space \(\ell^{n,m}_{p,q}\) of dimension \(mn\) with the unit ball \(B^{n,m}_{p,q}= \{x:|x|_{\ell^{n,m}_{p,q}}\leq 1\}\). We note that \(|\cdot|_{\ell^{n,m}_{p,p}}\equiv |\cdot|_{\ell^{mn}_p}\), \(1\leq p\leq\infty\). Recall that the Kolmogorov \(n\)-width of a subset \(V\) in a Banach space \(X\) is the value \[ d_n(V,X)= \inf_{L_n} \sup_{x\in V} \inf_{y\in L_n}|x-y|_X, \] where the outer infimum is taken over all subspaces \(L_n\) of \(X\), \(\dim L_n\leq n\). If now \(B^{n,m}_{p,q}\) is the unit ball in \(\ell^{n,m}_{p,q}\) then the main result of the present paper is Theorem 1. Let \(N=\left[{mn\over 2}\right]\). Then there exists an absolute positive constant \(c\) such that the following inequalities hold: \[ c\cdot{m\sqrt{\log\log m}\over\log m}\leq d_N(B^{n,m}_{1,\infty},\ell^{n,m}_{2,1})\leq m. \]
finite-dimensional spaces with mixed norms
scientific article; zbMATH DE number 796762