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Beschreibung:
The thesis at hand is concerned with the problem of sparse frequency estimation. It can be described as follows: Presented with a finite number of samples of an exponential sum, one wishes to calculate its spectrum, which is a discrete set. The focus is on the higher dimensional case, which attracted considerable attention in the last few years. In the first part of this thesis, we prove that in the one and two dimensional case, the sparse frequency problem is conditionally well-posed. More precisely, we give rather sharp estimates, which guarantee that if two exponential sums have well separated frequencies and their samples are close, so are their frequencies. Further, we give a posteriori error estimates. To prove that, we rely on special band-limited functions, satisfying certain sign patterns. And while such functions are known for quite some time, non of them satisfies an additional property we need. Therefore, we give a construction of such a function and start by reviewing the necessary results from sampling theory. In the second part, we turn to algorithms to actually solve the estimation problem. We quickly review classical univariate methods and then turn to so-called projection based methods. They cleverly reduce the higher dimensional problem to multiple one dimensional ones, by sampling the exponential sum along several lines. We give recovery guarantees for scattered as well as for parallel lines. For the latter case, we propose a new ESPRIT-like algorithm, combining the estimates along several lines into a single step. Finally, we turn to other multivariate methods. By explicitly considering the signal space, we can quite naturally deduce higher dimensional analogs of Prony's method, ESPRIT and MUSIC. That allows us to extend Sauer's sampling set, originally proposed for Prony's method, to ESPRIT and MUSIC, which reduces the number of necessary samples as well as the computational complexity significantly. ; Die vorliegende Arbeit befasst sich mit Frequenzschätzung von Exponentialsummen. Kurz ...