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The Sobolev space $H^{\varsigma}(\mathbb{R}^{d})$, where $\varsigma > d/2$,is an important function space that has many applications in various areas ofresearch. Attributed to the inertia of a measurement instrument, it isdesirable 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 $d/2<s<\varsigma$, weinvestigate the problem of constructing the approximations to all the functionsin $H^{\varsigma}(\mathbb{R}^{d})$ by nonuniform sampling. We first establishthe convergence rate of the framelet series in $(H^{s}(\mathbb{R}^{d}),H^{-s}(\mathbb{R}^{d}))$, and then construct the framelet approximationoperator that acts on the entire space $H^{\varsigma}(\mathbb{R}^{d})$. Weexamine the stability property for the framelet approximation operator withrespect to the perturbations of shift parameters, and obtain an estimate boundfor 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 inSobolev 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 theapproximation for every function in $H^{\varsigma}(\mathbb{R}^{d})$ bynonuniform sampling. In particular, the approximation error is robust to thejittering of the samples.

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