SUPERRESOLUTION, THE RECOVERY OF MISSING SAMPLES, AND VANDERMONDE MATRICES ON THE UNIT CIRCLE

The purpose of this paper is to study the conditioning of complex Vandermonde matrices, in reference to applications such as superresolution and the problem of recovering missing samples in band-limited signals. The results include bounds for the singular values of Vandermonde matrices whose nodes are complex numbers on the unit circle. It is shown that, under certain conditions, such matrices can be quite well-conditioned, contrarily to what happens in the real case. 1. INTRODUCTION Vandermonde matrices with real nodes are generally believed to be ill-conditioned, a reputation that they certainly deserve, at least when the nodes are real [5]. In fact, the condition number of real Vandermonde matrices has been shown to growth exponentially with the matrix order, at least for positive or symmetric nodes (and there is a conjecture that such node distribution is optimal). That is not always the case when the nodes of the matrix are allowed to be complex. There are Vandermonde matrices wi...