21,806 research outputs found
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
A theoretical basis for the Harmonic Balance Method
The Harmonic Balance method provides a heuristic approach for finding
truncated Fourier series as an approximation to the periodic solutions of
ordinary differential equations. Another natural way for obtaining these type
of approximations consists in applying numerical methods. In this paper we
recover the pioneering results of Stokes and Urabe that provide a theoretical
basis for proving that near these truncated series, whatever is the way they
have been obtained, there are actual periodic solutions of the equation. We
will restrict our attention to one-dimensional non-autonomous ordinary
differential equations and we apply the results obtained to a couple of
concrete examples coming from planar autonomous systems
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