ON THE OPTIMAL RECOVERY OF FUNCTIONS FROM THE CLASS W^{r,α}_2

Abstract

In this paper, the problem of optimal recovery of functions from the anisotropic Sobolev class W^r,α₂ in the power-logarithmic scale by the values of linear functionals is solved in the Hilbert metric, and the limiting error of the optimal computing unit is found. Thus, the following results are obtained here: 1) The exact order of error of optimal recovery of functions f∈ W^r,α₂ in by computing units constructed based on the values of linear functionals defined on the class under consideration has been established;2) The computing unit that realizes the established exact order is written out in explicit form; 3) The limiting error of the specified optimal сomputing unit is found, which preserves its optimality and can not be improved in order. The actuality of the problem studied here is that, firstly, the class W^r,α₂ is a finer scale of classifications of periodic functions by the rate of decrease of their trigonometric Fourier coefficients than the anisotropic Sobolev class in the power scale W^r₂ secondly, the set of computing units (l^(N), φ_(N)) with linear functionals is a fairly wide set containing all partial sums of Fourier series over all possible orthonormal systems, all possible finite convolutions with special kernels, as well as all finite sums of approximation used in orthowidths, linear widths and greedy algorithms.