On the Accuracy of Solving Confluent Prony Systems

Batenkov, Dmitry; Yomdin, Yosef

doi:10.1137/110836584

Cited by 45 publications

(72 citation statements)

References 42 publications

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“…We have m 2 ≤ 2 1 = 2 for the periodization x (2) and hence can apply the second case for the reconstruction of x (3) . We set L 2 = 1 and µ (2) = 0 for the first support index of x (2) .…”

Section: Reconstructing Real Nonnegative Vectors From Fourier Datamentioning

confidence: 99%

Deterministic sparse FFT algorithms

Wannenwetsch

2015

Proc Appl Math and Mech

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In this paper we consider sparse signals ${\bf x}\in {\bf C}^N$ which are known to vanish outside a support interval of length bounded by m < N. For the case that m is known, we propose a deterministic algorithm of complexity ${\cal O}(m\log m)$ for reconstruction of x from its discrete Fourier transform $\widehat{\bf x}\in {\bf C}^N.$ (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Section: Reconstructing Real Nonnegative Vectors From Fourier Datamentioning

confidence: 99%

Deterministic sparse FFT algorithms

Wannenwetsch

2015

Proc Appl Math and Mech

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“…Now consider the following well-known representation of M r as a product of three matrices (see e.g. [7]):…”

Section: Real Prony Space and Hyperbolic Polynomialsmentioning

confidence: 99%

“…Many research efforts are devoted to this task (see e.g. [3,7,10,17,18,20] and references therein). A basic question here is the following.…”

Section: Introductionmentioning

confidence: 99%

Geometry and Singularities of the Prony mapping

Batenkov¹,

Yomdin²

2014

jsing

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Abstract. Prony mapping provides the global solution of the Prony system of equations. This system appears in numerous theoretical and applied problems arising in Signal Reconstruction. The simplest example is the problem of reconstruction of linear combination of δ-functions of the form g(Global solution of the Prony system, i.e. inversion of the Prony mapping, encounters several types of singularities. One of the most important ones is a collision of some of the points x i . The investigation of this type of singularities has been started in [21] where the role of finite differences was demonstrated.In the present paper we study this and other types of singularities of the Prony mapping, and describe its global geometry. We show, in particular, close connections of the Prony mapping with the "Vieta mapping" expressing the coefficients of a polynomial through its roots, and with hyperbolic polynomials and "Vandermonde mapping" studied by V. Arnold.

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“…The most explored case is η = 1, λ k h(λ k z), λ k ∈ C, see [6,8,9,13,14,16,22]. The paper [13] contains several remarks 1 on the general case (3). Among other things, our motivation for studying (1) is based on that one needs to perform notably less arithmetic operations while computing (1), the amplitude and frequency sums with equal amplitudes, than in the case of (2) and (4), although all these sums have similar approximative properties with respect to the number of free parameters, as will be shown below.…”

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confidence: 99%