Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements

TL;DR

This paper establishes explicit density conditions for nonuniform sampling of multivariate bandlimited functions using derivatives or bunched measurements, with bounds that grow linearly with the number of derivatives or bunch size, and includes perturbation results.

Contribution

It provides new explicit density bounds for nonuniform sampling with derivatives and bunched measurements, extending classical sampling theory to multivariate and nonuniform contexts.

Findings

Density bounds grow linearly with the number of derivatives or bunch size.

Explicit frame bounds are derived for the sampling sets.

Perturbation results show stability under small deviations from ideal sampling sets.

Abstract

We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first $k$ derivatives. Along with explicit estimates of corresponding frame bounds, we derive the explicit density bound and show that, as $k$ increases, it grows linearly in $k+1$ with the constant of proportionality $1/\mathrm{e}$. Seeking larger gap conditions, we also prove a multivariate perturbation result for nonuniform samples that are sufficiently close to sets of sampling, e.g. to uniform samples taken at $k+1$ times the Nyquist rate. Additionally, in the univariate setting, we consider a related problem of so-called nonuniform bunched sampling, where in each sampling interval $s+1$ bunched measurements of a function are taken and the sampling intervals are permitted to be of different length. We derive an explicit density condition which grows linearly in $s+1$ for large $s$, with the constant of proportionality depending on the width of the bunches. The width of the bunches is allowed to be arbitrarily small, and moreover, for sufficiently narrow bunches and sufficiently large $s$, we obtain the same result as in the case of univariate sampling with $s$ derivatives.