Nonuniform Sampling and Recovery of Bandlimited Functions in Higher Dimensions

Abstract

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 ${\left(e^{-i⟨x_j,\cdot ⟩}\right)}_{j\in \mathbb{N}}$ form a Riesz basis for the $L_2$ space of a convex body containing the ball. Recovery results as well as corresponding approximation orders in terms of the parameter $\alpha$ are obtained.

17 pages. Submitted

References in corpus (2)

• Combining Riesz bases in $R^d$
• Approximation Rates for Interpolation of Sobolev Functions via Gaussians and Allied Functions

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