18,447 research outputs found
Laguerre-Angelesco multiple orthogonal polynomials on an -star
We investigate the type I and type II multiple orthogonal polynomials on an
-star with weight function , with . Each
measure , for , is supported on the semi-infinite
interval with . For both the type
I and the type II polynomials we give explicit expressions, the coefficients in
the recurrence relation, the differential equation and we obtain the asymptotic
zero distribution of the polynomials on the diagonal. Also, we give the
connection between the Laguerre-Angelesco polynomials and the Jacobi-Angelesco
polynomials on an -star.Comment: 33 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1804.0751
Asymptotic zero distribution of Jacobi-Pi\~neiro and multiple Laguerre polynomials
We give the asymptotic distribution of the zeros of Jacobi-Pi\~neiro
polynomials and multiple Laguerre polynomials of the first kind. We use the
nearest neighbor recurrence relations for these polynomials and a recent result
on the ratio asymptotics of multiple orthogonal polynomials. We show how these
asymptotic zero distributions are related to the Fuss-Catalan distribution.Comment: 19 pages, 2 figures. Some minor corrections and four new references
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Average Characteristic Polynomials of Determinantal Point Processes
We investigate the average characteristic polynomial where the 's are real random variables
which form a determinantal point process associated to a bounded projection
operator. For a subclass of point processes, which contains Orthogonal
Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a
sufficient condition for its limiting zero distribution to match with the
limiting distribution of the random variables, almost surely, as goes to
infinity. Moreover, such a condition turns out to be sufficient to strengthen
the mean convergence to the almost sure one for the moments of the empirical
measure associated to the determinantal point process, a fact of independent
interest. As an application, we obtain from a theorem of Kuijlaars and Van
Assche a unified way to describe the almost sure convergence for classical
Orthogonal Polynomial Ensembles. As another application, we obtain from
Voiculescu's theorems the limiting zero distribution for multiple Hermite and
multiple Laguerre polynomials, expressed in terms of free convolutions of
classical distributions with atomic measures.Comment: 26 page
Computing GCRDs of Approximate Differential Polynomials
Differential (Ore) type polynomials with approximate polynomial coefficients
are introduced. These provide a useful representation of approximate
differential operators with a strong algebraic structure, which has been used
successfully in the exact, symbolic, setting. We then present an algorithm for
the approximate Greatest Common Right Divisor (GCRD) of two approximate
differential polynomials, which intuitively is the differential operator whose
solutions are those common to the two inputs operators. More formally, given
approximate differential polynomials and , we show how to find "nearby"
polynomials and which have a non-trivial GCRD.
Here "nearby" is under a suitably defined norm. The algorithm is a
generalization of the SVD-based method of Corless et al. (1995) for the
approximate GCD of regular polynomials. We work on an appropriately
"linearized" differential Sylvester matrix, to which we apply a block SVD. The
algorithm has been implemented in Maple and a demonstration of its robustness
is presented.Comment: To appear, Workshop on Symbolic-Numeric Computing (SNC'14) July 201
Discrete integrable systems generated by Hermite-Pad\'e approximants
We consider Hermite-Pad\'e approximants in the framework of discrete
integrable systems defined on the lattice . We show that the
concept of multiple orthogonality is intimately related to the Lax
representations for the entries of the nearest neighbor recurrence relations
and it thus gives rise to a discrete integrable system. We show that the
converse statement is also true. More precisely, given the discrete integrable
system in question there exists a perfect system of two functions, i.e., a
system for which the entire table of Hermite-Pad\'e approximants exists. In
addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page
Crystallization of random trigonometric polynomials
We give a precise measure of the rate at which repeated differentiation of a
random trigonometric polynomial causes the roots of the function to approach
equal spacing. This can be viewed as a toy model of crystallization in one
dimension. In particular we determine the asymptotics of the distribution of
the roots around the crystalline configuration and find that the distribution
is not Gaussian.Comment: 10 pages, 3 figure
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
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