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Local and Global Geometry of Prony Systems and Fourier Reconstruction of Piecewise-Smooth Functions

  • D. Batenkov
  • Y. Yomdin
Conference paper
Part of the Abel Symposia book series (ABEL, volume 9)

Abstract

Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general perspective, addressing questions of global solvability and stable inversion. Of special interest are the so-called “near-singular” situations, such as a collision of two closely spaced nodes. We also discuss the problem of reconstructing piecewise-smooth functions from their Fourier coefficients, which is easily reduced by a well-known method of K. Eckhoff to solving a particular Prony system. As we show in the paper, it turns out that a modification of this highly nonlinear method can reconstruct the jump locations and magnitudes of such functions, as well as the pointwise values between the jumps, with the maximal possible accuracy.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer ShevaIsrael
  2. 2.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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