Nonuniform sampling and approximation in Sobolev space from the perturbation of framelet system

Abstract

The Sobolev space $H^{\varsigma}(\mathbb{R}^{d})$, where $\varsigma > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in sampling theory to recover a function by its nonuniform sampling. In the present paper, based on dual framelet systems for the Sobolev space pair $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$, where $d/2<s<\varsigma$, we investigate the problem of constructing the approximations to all the functions in $H^{\varsigma}(\mathbb{R}^{d})$ by nonuniform sampling. We first establish the convergence rate of the framelet series in $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$, and then construct the framelet approximation operator that acts on the entire space $H^{\varsigma}(\mathbb{R}^{d})$. We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters, and obtain an estimate bound for the perturbation error. Our result shows that under the condition $d/2<s<\varsigma$, the approximation operator is robust to shift perturbations. Motivated by some recent work on nonuniform sampling and approximation in Sobolev space (e.g., [20]), we don't require the perturbation sequence to be in $\ell^{\alpha}(\mathbb{Z}^{d})$. Our results allow us to establish the approximation for every function in $H^{\varsigma}(\mathbb{R}^{d})$ by nonuniform sampling. In particular, the approximation error is robust to the jittering of the samples.