A scale and shift paradigm for sparse interpolation in one and more dimensions

Abstract

Sparse interpolation from at least 2n uniformly spaced interpolation points tj can be traced back to the exponential fitting method [MATH HERE] of de Prony from the 18-th century [5]. Almost 200 years later this basic problem is also reformulated as a generalized eigenvalue problem [8]. We generalize (1) to sparse interpolation problems of the form [MATH HERE] and some multivariate formulations thereof, from corresponding regular interpolation point patterns. Concurrently we introduce the wavelet inspired paradigm of dilation and translation for the analysis (2) of these complex-valued structured univariate or multivariate samples. The new method is the result of a search on how to solve ambiguity problems in exponential analysis, such as aliasing which arises from too coarsely sampled data, or collisions which may occur when handling projected data.