  The Sobolev space $H^{s}(\mathbb{R}^{d})$, where $s > d/2$, is an important
function space that has many applications in various areas of research.
Attributed to the inertia of a measuring instrument, it is desirable in
sampling theory to reconstruct a function by its nonuniform samples. In the
present paper, we investigate the problem of constructing the approximation to
all the functions in $H^{s}(\mathbb{R}^{d})$ with nonuniform samples by
utilizing dual framelet systems for the Sobolev space pair
$(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$. We first establish the
convergence rates of the framelet series in $(H^{s}(\mathbb{R}^{d}),
H^{-s}(\mathbb{R}^{d}))$, and then construct the framelet approximation
operator holding for the entire space $H^{s}(\mathbb{R}^{d})$. Using the
approximation operator, any function in $H^{s}(\mathbb{R}^{d})$ can be
approximated at the exponential rate with respect to the scale level. We
examine the stability property for the perturbations of the framelet
approximation operator with respect to shift parameters, and obtain an estimate
bound for the perturbation error. Our result shows that under the condition $s
> d/2$, the approximation operator is robust to the shift perturbation. These
results are used to establish the nonuniform sampling approximation for every
function in $H^{s}(\mathbb{R}^{d})$. In particular, the new nonuniform sampling
approximation error is robust to the jittering of the samples.
