Error analysis of the MMSE estimator for multidimensional band-limited extrapolations from finite samples

The problem of reconstructing a multidimensional band-limited signal based on a finite set of samples has been considered by many researchers. For this problem, the minimum mean-squared error (MMSE) estimator of the form of a sum of weighted band-limited interpolating functions that are identical in shape but are centered at the irregularly spaced sample points was derived by D.S. Chen and J.P. Allebach. It was also proved that the estimator is identical to the well-known minimum-energy band-limited interpolator. In this paper, what we are interested in is the extrapolation problem from finite samples. In this case, the sample points are restricted in a bounded spatial domain. We prove that when the sampling density increases in each dimension of the bounded spatial domain, under certain conditions the MMSE estimator exponentially converges to the original signal and uniformly in any bounded spatial domain. Therefore, the MMSE estimator can be thought of as an extrapolator. Also, the Tikhonov regularization is employed to deal with the case of distorted data. Error estimates and numerical examples are presented to illustrate the performance.