Some lower bounds for optimal sampling recovery of functions with mixed smoothness

TL;DR

This paper investigates the fundamental limits of nonlinear sampling recovery for functions with mixed smoothness, providing new lower bounds where existing methods for linear recovery do not apply.

Contribution

It introduces a novel approach to establish lower bounds for nonlinear sampling recovery, addressing a gap in the theoretical understanding of optimal rates.

Findings

Established new lower bounds for nonlinear sampling recovery

Demonstrated limitations of existing techniques for nonlinear cases

Provided insights into the differences between linear and nonlinear recovery bounds

Abstract

Recently, there was a substantial progress in the problem of sampling recovery on function classes with mixed smoothness. Mostly, it has been done by proving new and sometimes optimal upper bounds for both linear sampling recovery and for nonlinear sampling recovery. In this paper we address the problem of lower bounds for the optimal rates of nonlinear sampling recovery. In the case of linear recovery one can use the well developed theory of estimating the Kolmogorov and linear widths for establishing some lower bounds for the optimal rates. In the case of nonlinear recovery we cannot use the above approach. It seems like the only technique, which is available now, is based on some simple observations. We demonstrate how these observations can be used.