204 research outputs found
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
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
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