Exponential Approximation of Multivariate Bandlimited Functions from Average Oversampling

TL;DR

This paper introduces an explicit method for reconstructing multivariate bandlimited functions from finite average oversampling data, achieving exponentially decaying approximation errors, which enhances stability and practicality over traditional sampling methods.

Contribution

It provides a novel explicit reconstruction technique with optimal decay of Fourier transform, improving stability and efficiency in average oversampling of multivariate functions.

Findings

Reconstruction error decays exponentially with data size.

Method improves stability over point sampling.

Applicable to multivariate bandlimited functions.

Abstract

Instead of sampling a function at a single point, average sampling takes the weighted sum of function values around the point. Such a sampling strategy is more practical and more stable. In this note, we present an explicit method with an exponentially-decaying approximation error to reconstruct a multivariate bandlimited function from its finite average oversampling data. The key problem in our analysis is how to extend a function so that its Fourier transform decays at an optimal rate to zero at infinity.