Sensitivity of best recovery in the Sobolev spaces $W_\infty ^{r,d}$, $\widetilde W_\infty ^{r,d}$for perturbed sampling

Abstract

Let $I^d$be the d‐dimensional cube, $I^d = [0,1]^d$, and let $F \ni f \mapsto Sf \in L_\infty (I^d )$be a linear operator acting on the Sobolev space F, where Fis either

$$$$

or

$$$$

where

$$\left\| f \right\|_F = \sum\limits_{\left| m \right| = r} {\mathop {{\text{esssup}}}\limits_{x \in I^d } \left| {\frac{{\partial f^{\left| m \right|} }} {{\partial x_1^{m_1 } \partial x_2^{m_2 } \cdot \cdot \cdot \partial x_d^{m_d } }}(x)} \right|.} $$

We assume that the problem elements fsatisfy the condition $\sum\nolimits_{\left| m \right| = r} {{\text{esssup}}} _{x \in I^d } \left| {f^{(m)} (x)} \right| \leqslant 1$and that Sis continuous with respect to the supremum norm.

We study sensitivity of optimal recovery of Sfrom inexact samples of ftaken at npoints forming a uniform grid on $I^d$. We assume that the inaccuracy in reading the sample vector is measured in the pth norm and bounded by a nonnegative number δ. The sensitivity is defined by the difference between the optimal errors corresponding to the exact and perturbed readings, respectively. Our main result is that this difference is bounded by $\mathcal{A}\delta$, where $\mathcal{A}$is a positive constant independent of the number of samples. This indicates that the curse of dimension, which badly affects the optimal errors, does not extend to sensitivity.