\(n\)-widths of \(H^ p\)-spaces in \(L_ p(-1,1)\)
Kolmogorov \(n\)-widths \(d_ n\), Gel'fand \(n\)-widths \(d^ n\), and linear \(n\)-widths \(\delta_ n\) of the Hardy spaces \(H^ p\) in \(L_ q(-1,1)\) are estimated. For \(1\leq q< p\leq \infty\) and some constants \(k_ 1\), \(k_ 2\) the inequality \[ \begin{aligned} k_ 1 & n^{(1/2)(1/q-1/p)}\exp\left(- \pi\sqrt{{n\over 2}\Bigl({1\over q}-{1\over p}\Bigr)}\right)\\ & \leq d_ n(H^ p,L_ q(-1, 1)),\;d^ n(H^ p, L_ q(-1,1)),\;\delta_ n(H^ p, L_ q(-1,1))\\ & \leq k_ 2 n^{1/2q} \exp\left(- \pi \sqrt{{n\over 2}\Bigl({1\over q}- {1\over p}\Bigr)}\right).\end{aligned} \] is shown among others.
scientific article; zbMATH DE number 97385