Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness

TL;DR

This paper investigates the asymptotic properties of function classes with small mixed smoothness, providing bounds on Kolmogorov widths and implications for optimal sampling recovery in the L2 norm.

Contribution

It establishes new bounds on Kolmogorov widths for classes with small mixed smoothness and links these bounds to improved sampling recovery estimates.

Findings

Derived bounds for Kolmogorov widths of classes with small mixed smoothness.

Established new upper bounds for optimal sampling recovery in L2 norm.

Connected Kolmogorov width estimates to practical sampling strategies.

Abstract

Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.