Optimal C(N)D-recovery of Functions from the Anisotropic Generalized Sobolev Class

Abstract

In this paper, the problem of recovering functions from an anisotropic generalized Sobolev class in the context of the computational (numerical) diameter is completely solved. The anisotropic generalized Sobolev class $W_2^{\omega_{r_1}, \ldots, \omega_{r_s}}$, which arises from the classification of periodic functions according to the rate of decay of their trigonometric Fourier coefficients, represents a finer scale of function characteristics than the anisotropic Sobolev class
$W_2^{r_1, \ldots, r_s}$ in the power scale. The paper presents two-sided estimates, up to multiplicative constants, for the approximation error of functions from the considered class by computational aggregates constructed from information given in the form of linear functionals.
In addition, based on trigonometric Fourier coefficients, the optimal computational aggregate is explicitly constructed. It is noted that the set of computational aggregates $(l^{(N)}, \varphi_N)$ used in the lower bounds is sufficiently wide, containing all partial sums of Fourier series with respect to various orthonormal systems, all possible finite convolutions with special kernels, as well as all finite approximation sums used in orthogonal widths, linear widths, and greedy algorithms. Furthermore, in the second part of the paper, the limiting error $\overline{\varepsilon}_N$ of numerical information in the form of trigonometric Fourier coefficients is obtained, which preserves the optimality of the computational aggregate and cannot be improved in order. In the third part, it is proved that all computational aggregates constructed using trigonometric Fourier coefficients do not have a limiting error greater than $\overline{\varepsilon}_N$.