Approximation of functions on a grid (Q2387089)

The author deals with a discrete analog of the Kolmogorov problem concerning the order of approximation in \(W^r\). He shows, among others, that the asymptotic properties of \[ C_m(W^r_q)=\sup_{f\in W^r_q}\| f-S_{m-1}(f;x_n)\| , \] for \(m,q\to\infty\), depend on the limit of the ratio of the order of the Fourier sums and the number of points of the homogeneous grid. In particular, they depend on the rationality or irrationality of this limit. It was found that unlike the continuous case, this situation is naturally related to the Riemann type functions.