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Let $G$ be a finite union of disjoint and bounded Jordan domains in the complex plane, let $mathcal{K}$ be a compact subset of $G$ and consider the set $G^star$ obtained from $G$ by removing $mathcal{K}$; i.e., $G^star:=Gsetminus mathcal{K}$. We refer to $G$ as an archipelago and $G^star$ as an archipelago with lakes. Denote by ${p_n(G,z)}_{n=0}^infty$ and ${p_n(G^star,z)}_{n=0}^infty$, the sequences of the Bergman polynomials associated with $G$ and $G^star$, respectively; that is, the orthonormal polynomials with respect to the area measure on $G$ and $G^star$. The purpose of the paper is to show that $p_n(G,z)$ and $p_n(G^star,z)$ have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for $G^star$ are determined by the boundary of $G$. As a consequence we can analyze certain asymptotic properties of $p_n(G^star,z)$ by using the corresponding results for $p_n(G,z)$, which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.
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We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the equilibrium measure, and we compute their Jacobi matrices via standard procedures, suitably enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, that show stability and efficiency of the whole procedure. As a secondary result, we also compute Jacobi matrices of equilibrium measures on finite sets of intervals, and of balanced measures of Iterated Function Systems. These algorithms can reach large orders: we study the asymptotic behavior of the orthogonal polynomials and we show that they can be used to efficiently compute Greens functions and conformal mappings of interest in constructive function theory.
We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are equivalent under suitable assumptions. We also show relations between these measures and the classical element-wise and norm-wise backward error measures.
In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product begin{equation*} leftlangle p,qrightrangle _{s}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{prime }(0)q^{prime }(0), end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.
For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.
This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure $mu$, i.e., begin{equation*} frac{1}{(x-c)}dmu (x)+Ndelta (x-c), end{equation*} for some free parameter $N in mathbb{R}_{+}$ and shift $c$. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass $N$ tends to infinity as well as we characterize the precise values of $N$ such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure $mu$. When $mu$ is semi-classical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation $c$ is located outside of the support of $mu$.
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