Comptes rendus mathématiques de l'Académie des sciences
Mathematical Reports of the Academy of Science

Volume		Number		The Royal Society of Canada	December
Tom	29	Numéro	4	La Société royale du Canada	Decembre	2007

N. SARIG and Y. YOMDIN Signal Acquisition from Measurements via Non-Linear Models

ABSTRACT. We consider the problem of reconstruction of a non-linear finite-parametric model M = M_p(x) with p = (p₁,..., p_r) a set of parameters, from a set of measurements m_j(M) . In this paper m_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<sub>p</sub>(x) into the measurements m<sub>j</sub> and compute explicitly the resulting ``symbolic" expressions of m_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 in the parameters p , which we then try to solve. The aim of this paper is to review some recent results

RÉSUMÉ. Nous étudions le problème de reconstruction d'un modèle non-linéaire parametrisé M = M_p(x) ,

V. A. KISUNKO Cauchy Type Integrals and a D -Moment Problem

ABSTRACT. We consider a Cauchy-type integral F(z) = $\int_{{\Gamma}}^{}$${\frac{{g(\xi)\,\d\xi}}{{\xi-z}}}$ , where g is a piecewise analytic function satisfying an n -th order linear homogeneous differential equation Ly = ${\frac{{\d ^n y}}{{\d z^n}}}$ + c_n-1${\frac{{\d ^{n-1}}}{{\d z^{n-1}}}}$ + ... + c₀y = 0 with coefficients c_k $\in$ $\mathbb {C}$(z) rational functions. Our main theorem asserts that the function F satisfies a linear non-homogeneous equation Ly = R with R a rational function. The precise description of R leads to the solution of a vanishing problem and to the solution of a moment-type problem, which we call D-moment problem.

RÉSUMÉ. On considère une integrale du type Cauchy F(z) = $\int_{{\Gamma}}^{}$${\frac{{g(\xi)\d\xi}}{{\xi-z}}}$ , où g est une fonction analytique par morceaux satisfaisant une équation différentielle linéaire homogène d'ordre n , Ly = ${\frac{{\d ^n y}}{{\d z^n}}}$ + c_n-1${\frac{{\d ^{n-1}}}{{\d z^{n-1}}}}$ + ... + c₀y = 0 , aux coefficients c_k $\in$ $\mathbb {C}$(z)

ROBIN J. DEELEY The Range of the Orbit Operator and Invariant Subspaces

ABSTRACT. To a bounded linear operator and a vector in the Hilbert space on which it acts we associate a linear map which we call the orbit operator. We prove a number of results linking properties of the range of the orbit operator to the existence of invariant subspaces of the original operator.

RÉSUMÉ.