Nonuniform Sampling and Recovery of Bandlimited Functions in Higher Dimensions

We provide sufficient conditions on a family of functions $({\varphi }_{\alpha }{)}_{\alpha \in A}:{\mathbb{R}}^d\to \mathbb{R}$ for sampling of multivariate bandlimited functions at certain nonuniform sequences of points in ${\mathbb{R}}^d$. We consider interpolation of functions whose Fourier transform is supported in some small ball in ${\mathbb{R}}^d$ at scattered points $(x_j{)}_{j\in \mathbb{N}}$ such that the complex exponentials $\left(e^{-i\langle x_j,\cdot\rangle}\right)_{j\in$\mathbb{N}$}$formaRieszbasisforthe$L_2$ space of a convex body containing the ball. Recovery results as well as corresponding approximation orders in terms of the parameter $\alpha$ are obtained.