On trigonometric n-widths and their generalization
The author proves that the subspaces of the form \(span\{e^{ik_ 1.},...,e^{ik_ n.}\}\), with certain \(k_ 1,...,k_ n\in {\mathbb{Z}}\), are asymptotically optimal for the n-widths of the Sobolev classes \(W^ r_ p\) in \(L_ q (1\leq p\leq 2\leq q<\infty\), \(r>p^{-1}-q^{-1})\). For \(r>1/p+1/2\) this fact was established by \textit{V. E. Maiorov} [Math. USSR, Sb. 41, 361-382 (1982; Zbl 0478.41017)]. The proof is based on a probability argument and is derived from a more general statement for functions representable as convolutions on a compact Abelian group.
scientific article; zbMATH DE number 3863798