Criterion for the asymptotic optimality of projection-grid subspaces (Q1316927)

The author deals with approximation in the Sobolev-type space \(H_ s\) endowed with the norm \[ \| u\|_ s= \left\{ \int_{E_ n} |\widehat u(\xi)|^ 2(1+ |\xi|^ 2)^ sd\xi\right\}^{1/2}, \] where \(\widehat u\) is the Fourier image of \(u\in H_ s\). By means of a given lattice \(\Lambda\) in the \(n\)-dimensional Euclidean space \(E_ n\) (i.e. a set consisting of all the points of the form \(\alpha_ 1 f_ 1+\cdots+ \alpha_ n f_ n\), where \(f_ 1,\dots,f_ n\) are \(n\) linearly independent points in \(E_ n\) and \(\alpha_ 1,\dots,\alpha_ n\) are arbitrary integers), the author associates with each function \(\varphi\in H_ s\) a family \(\{B(\Lambda^ h,\varphi^ h)\mid h\in (0,1]\}\) of closed linear subspaces of \(H_ s\). The main theorem of the paper characterizes, in terms of the Fourier image, those functions \(\varphi\in H_ s\) for which \[ \limsup_{h\to 0} \chi(\Lambda,\varphi,h)/h^{m-s}\leq K<\infty, \] where \(m\) is a real number satisfying \(s< m\) and \(\chi(\Lambda,\varphi,h)\) is defined by \[ \chi(\Lambda,\varphi,h)=\sup_{\| u\|_ m=1}\inf_{w\in B(\Lambda^ h,\varphi^ h)} \| u- w\|_ s. \] {}.