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

The Sobolev space Hς (ℝd), where ς > 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 (Hs(ℝd), H−s(ℝd)), where d/2 < s < ς, we investigate the problem of constructing the approximations to all the functions in Hς(ℝd) by nonuniform sampling. We first establish the convergence rate of the framelet series in (Hs(ℝd), H−s(ℝd)), and then construct the framelet approximation operator that acts on the entire space Hς (ℝd). 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 < ς, the approximation operator is robust to shift perturbations. Motivated by Hamm (2015)’s work on nonuniform sampling and approximation in the Sobolev space, we do not require the perturbation sequence to be in ℓα(ℤd). Our results allow us to establish the approximation for every function in Hς(ℝd) by nonuniform sampling. In particular, the approximation error is robust to the jittering of the samples.