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Direct spreading measures of Laguerre polynomials  [PDF]
P. Sánchez-Moreno,D. Manzano,J. S. Dehesa
Mathematics , 2010, DOI: 10.1016/j.cam.2010.07.022
Abstract: The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean-square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q is a natural number) is also found in terms of the polynomials parameters by means of two error-free computing approaches; one makes use of the Lauricella functions, which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n>>1).
Spreading lengths of Hermite polynomials  [PDF]
P. Sánchez-Moreno,J. S. Dehesa,D. Manzano,R. Yá?ez
Physics , 2009, DOI: 10.1016/j.cam.2009.09.043
Abstract: The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, it is computationally proved the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed.
Orthogonal Polynomials with Respect to Self-Similar Measures  [PDF]
Steven M. Heilman,Philip Owrutsky,Robert S. Strichartz
Mathematics , 2009,
Abstract: We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theory of dynamical systems, for studying families $P_{n}(x)$ of orthogonal functions as functions of $n$ for fixed values of $x$.
Orthogonal polynomials of equilibrium measures supported on Cantor sets  [PDF]
Giorgio Mantica
Mathematics , 2013,
Abstract: 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 Green's functions and conformal mappings of interest in constructive function theory.
Remarks on orthogonal polynomials with respect to varying measures and related problems  [cached]
Xin Li
International Journal of Mathematics and Mathematical Sciences , 1993, DOI: 10.1155/s0161171293000298
Abstract: We point out the relation between the orthogonal polynomials with respect to (w.r.t.) varying measures and the so-called orthogonal rationals on the unit circle in the complex plane. This observation enables us to combine different techniques in the study of these polynomials and rationals. As an example, we present a simple and short proof for a known result on the weak-star convergence of orthogonal polynomials w.r.t, varying measures. Some related problems are also considered.
Measures for orthogonal polynomials with unbounded recurrence coefficients  [PDF]
A. I. Aptekarev,J. S. Geronimo
Mathematics , 2014,
Abstract: Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are asymptotics for the polynomials.
Orthogonal polynomials in several variables for measures with mass points  [PDF]
A. M. Delgado,L. Fernandez,T. E. Perez,M. A. Pinar,Y. Xu
Mathematics , 2009,
Abstract: Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials associated with $d\mu$, so are the reproducing kernels associated with these polynomials. The explicit formulas that are obtained are further specialized in the case of Jacobi measure on the simplex, with mass points added on the vertices, which are then used to study the asymptotics kernel functions for $d\nu$.
Muckenhoupt inequality with three measures and applications to Sobolev orthogonal polynomials  [PDF]
E. Colorado,D. Pestana,J. M. Rodriguez,E. Romera
Mathematics , 2012,
Abstract: We generalize the classical Muckenhoupt inequality with two measures to three under appropriate conditions. As a consequence, we prove a simple characterization of the undedness of the multiplication operator and thus of the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials, for a large class of measures which includes the most usual examples in the literature.
Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations  [PDF]
Holger Dette,Jens Wagener
Mathematics , 2009,
Abstract: We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szeg\"{o} recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [-1,1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.
Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials  [PDF]
José M. Rodríguez,José M. Sigarreta
Mathematics , 2008,
Abstract: In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.
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