arXiv subjects

Yosef Yomdin

Publications and source records attributed to Yosef Yomdin.

45 records · Page 3Linked to original sources

arXiv2012

Reconstruction of Planar Domains from Partial Integral Measurements

We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewise-algebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [Bat]* and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects. * D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse Problems 25 (2009) 105001

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arXiv2011

Algebraic reconstruction of piecewise-smooth functions from integral measurements

This paper presents some results on a well-known problem in Algebraic Signal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments, Fourier coefficients, Radon transform, etc.). Our results concern reconstruction (from the moments or Fourier coefficients) of signals in two specific classes: linear combinations of shifts of a given function, and "piecewise $D$-finite functions" which satisfy on each continuity interval a linear differential equation with polynomial coefficients. In each case the problem is reduced to a solution of a certain type of non-linear algebraic system of equations ("Prony-type system"). We recall some known methods for explicitly solving such systems in one variable, and provide extensions to some multi-dimensional cases. Finally, we investigate the local stability of solving the Prony-type systems.

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arXiv2011

Generalized Remez Inequality for $(s,p)$-Valent Functions

The classical Remez inequality bounds the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z$ of positive measure in $[-1,1]$. It was shown in \cite{Yom3} that the Lebesgue measure in the Remez inequality can be replaced by a certain geometric invariant $ω_d(Z)$ which can be effectively estimated in terms of the metric entropy of $Z$ and which may be nonzero for discrete and even finite sets $Z$. In the present paper we first obtain an essentially sharp Remez-type inequality in the spirit of \cite{Yom3} for complex polynomials of one variable, introducing metric invariants $c_d(Z)$ and $ø_{cd}(Z)$ for an arbitrary subset $Z\subset D_1\subset {\mathbb C}$. These invariants translate into the the metric language the classical Cartan lemma (see \cite{Gor} and references therein). Next we introduce $(s,p)$-valent functions, which provide a natural generalization of $p$-valent ones (see \cite{Hay} and references therein). We prove a "distortion theorem" for such functions, comparing them with polynomials sharing their zeroes. On this base we extend to $(s,p)$-valent functions our polynomial Remez-type inequality. As the main example we consider restrictions $g$ of polynomials of a growing degree to a fixed algebraic curve, for which we obtain an essentially sharp "local" Remez-type inequality, stressing the role of the geometry of singularities of $g$. Finally, we obtain for such functions $g$ a "global" Remez-type inequality which is valid for all the branches of $g$ and involves both the geometry of singularities of $g$ and its monodromy.

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arXiv2010

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 $k$ Fourier coefficients of a piecewise-$C^{2d+1}$ function, recovers the locations of the jumps with accuracy $\sim k^{-(d+2)}$, and the values of the function between the jumps with accuracy $\sim k^{-(d+1)}$ (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.

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arXiv2009

An "algebraic" reconstruction of piecewise-smooth functions from integral measurements

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arXiv2008

Signal Acquisition from Measurements via Non-Linear Models

We consider the problem of reconstruction of a non-linear finite-parametric model $M=M_p(x),$ with $p=(p_1,...,p_r)$ a set of parameters, from a set of measurements $μ_j(M)$. In this paper $μ_j(M)$ are always the moments $m_j(M)=\int x^jM_p(x)dx$. This problem is a central one in Signal Processing, Statistics, and in many other applications. We concentrate on a direct (and somewhat "naive") approach to the above problem: we simply substitute the model function $M_p(x)$ into the measurements $μ_j$ and compute explicitly the resulting "symbolic" expressions of $μ_j(M_p)$ in terms of the parameters $p$. Equating these "symbolic" expressions to the actual measurement results, we produce a system of nonlinear equations on the parameters $p$, which we consequently try to solve. The aim of this paper is to review some recent results (mostly of \cite{Vet5,Mil1,Mil2,Put1,Vet4,Vet3,Mil3,Mil4,Vet2}) in this direction, stressing the algebraic structure of the arising systems and mathematical tools required for their solutions. In particular, we discuss the relation of the reconstruction problem above with the recent results of \cite{bfy,bry,chr,pak1,pak2,pak3,pry,ry} on the vanishing problem of generalized polynomial moments and on the Cauchy-type integrals of algebraic functions. The accompanying paper \cite{Kis1} (this volume) provides a solution method for a wide class of reconstruction problems as above, based on the study of linear differential equations with rational coefficient, which are satisfied by the moment generating function of the problem.

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arXiv2007

Linear versus Non-linear Acquisition of Step-Functions

We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and non-linear approaches. As a model case we use piecewise-constant functions on [0,1], in particular, the Heaviside jump function. Considered as a curve in the Hilbert space, it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov's n-width and entropy, as well as Temlyakov's (N,m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other.

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arXiv2006

Rotation of Trajectories of Lipschitz Vector Fields

We prove that in finite time a trajectory of a Lipschitz vector field in $\hbox{\bbbb R}^{\hbox{\tmm n}}$ can not have infinite rotation around a given point. This result extends to the mutual rotation of two trajectories of a field in $\hbox{\bbbb R}^{\hbox{\tmm 3}}$: this rotation is bounded from above on any finite time interval. The bounds we give are only in terms of the Lipschitz constant of the field and the length of the time interval.

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arXiv2005

Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups

We prove various reconstruction theorems about open subsets of normed spaces. E.g. if the uniformly continuous homeomorphism groups of two such sets are isomorphic, then this isomorphism is induced by a uniformly continuous homeomorphism between these open sets.

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