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

TL;DR

This work provides nontrivial lower bounds for nonlinear sampling recovery on function classes with mixed smoothness, notably establishing $\varrho_m^{o}({\mathbf H}^r_q,L_p) \ge c(d)\,m^{-r+1/q-1/p}(\log m)^{(d-1)/p}$ for $1\le q\le p\le \infty$, $p>1$, $r>1/q$, and extending to related regimes and structural class settings. The authors develop the lower-bound machinery by constructing test functions from finite Fourier subspaces, Fejér and de la Vallée Poussin kernels, and leveraging hyperbolic-cross decompositions, then connect these to ${\mathbf W}^r_q$ and ${\mathbf H}^r_q$ via embeddings. They further provide lower bounds in the $1\le p\le q\le \infty$ regime and for classes with structural $A_\beta$-norms, revealing how logarithmic factors arise and how nonlinear sampling can outperform linear strategies in certain parameter ranges. The results extend the theory of optimal sampling recovery beyond $L_2$ and offer a unified framework for lower bounds across classical and structurally constrained mixed-smoothness classes, with implications for the design of nonlinear sampling schemes. Overall, the paper clarifies the limitations and scope of nonlinear recovery in high-dimensional harmonic-analytic settings and identifies key open questions for extending these 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.

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.1
• Theorem 1.1: KoTe
• Theorem 1.2: KoTe
• Theorem 1.3
• Proposition 1.2
• Theorem 2.1
• Theorem 2.2