Approximation by elements of a finite-dimensional subspace of functions from various Sobolev or Nikol'skij spaces (Q920321)

Let \(F^ R_ q\) stand either for the anisotropic Sobolev class or for the Nikolskij class \((SW^ R_{q,\alpha}\) or \(NH^ R_ q)\). The following problem is treated: Select a proper ordering of a trigonometric system \(\{e^{i(k,x)}\}\) in order that the partial sums of periodic functions should give good approximation of these functions. The quantity \[ d_ M^{\perp}(F^ R_ q)_ p=\inf_{\{u_ i\}^ M_{i=1}}\sup_{f\in F^ R_ q}\| f-\sum^{M}_{i=1}(f,u_ i)\cdot u_ i\|_ p, \] where the infimum is taken over all orthonormal systems of bounded functions, is shown to satisfy certain estimates. Given M natural and a parallelepiped P, the author finds a system \(\{u_ i\}^ N_{i=1}\) with the least possible N such that for all \(R\in P\) the inequality \[ \sup_{f\in F^ R_ q}\| f- \sum^{N}_{i=1}(f,u_ i)u_ i\|_ p\leq K.d_ M^{\perp}(F^ R_ q)_ p \] holds. An asymptotical estimate for this smallest N by means of M is given.