50 research outputs found
Geometry and Singularities of the Prony mapping
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 -functions of the form
, with the unknown parameters $a_{i},\
x_{i},\ i=1,...,n,m_{k}=\int x^{k}g(x)dx.x_{i}.$ The investigation of this type of
singularities has been started in \cite{yom2009Singularities} 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.Comment: arXiv admin note: text overlap with arXiv:1301.118
Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions
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.Comment: arXiv admin note: text overlap with arXiv:1211.068
Algebraic Fourier reconstruction of piecewise smooth functions
Accurate reconstruction of piecewise-smooth functions from a finite number of
Fourier coefficients is an important problem in various applications. The
inherent inaccuracy, in particular the Gibbs phenomenon, is being intensively
investigated during the last decades. Several nonlinear reconstruction methods
have been proposed, and it is by now well-established that the "classical"
convergence order can be completely restored up to the discontinuities. Still,
the maximal accuracy of determining the positions of these discontinuities
remains an open question. In this paper we prove that the locations of the
jumps (and subsequently the pointwise values of the function) can be
reconstructed with at least "half the classical accuracy". In particular, we
develop a constructive approximation procedure which, given the first
Fourier coefficients of a piecewise- function, recovers the locations
of the jumps with accuracy , and the values of the function
between the jumps with accuracy (similar estimates are
obtained for the associated jump magnitudes). A key ingredient of the algorithm
is to start with the case of a single discontinuity, where a modified version
of one of the existing algebraic methods (due to K.Eckhoff) may be applied. It
turns out that the additional orders of smoothness produce a highly correlated
error terms in the Fourier coefficients, which eventually cancel out in the
corresponding algebraic equations. To handle more than one jump, we propose to
apply a localization procedure via a convolution in the Fourier domain