Asymptotic zero distribution for a class of multiple orthogonal polynomials

Coussement, E.; Coussement, J.; Assche, Walter Van

doi:10.1090/s0002-9947-08-04535-2

Cited by 43 publications

(68 citation statements)

References 30 publications

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“…Therefore, Γ 0 ∪ Γ 1 = (−∞, 1]. The density (6.2) was already given in [3] and (6.3) follows in a similar way.…”

Section: 2supporting

confidence: 52%

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An Equilibrium Problem for the Limiting Eigenvalue Distribution of Banded Toeplitz Matrices

Duits¹,

Kuijlaars²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. We study the limiting eigenvalue distribution of n×n banded Toeplitz matrices as n → ∞. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as n → ∞. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.

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“…Therefore, Γ 0 ∪ Γ 1 = (−∞, 1]. The density (6.2) was already given in [3] and (6.3) follows in a similar way.…”

Section: 2supporting

confidence: 52%

“…So we obtain two contours Γ 0 and Γ 1 with two associated measures µ 0 and µ 1 . This example appeared in [3], in which the authors gave explicit expressions for Γ 0 and µ 0 . The following proposition also contains expressions for Γ 1 and µ 1 .…”

Section: 2mentioning

confidence: 99%

An Equilibrium Problem for the Limiting Eigenvalue Distribution of Banded Toeplitz Matrices

Duits¹,

Kuijlaars²

2008

SIAM J. Matrix Anal. & Appl.

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show abstract

“…The average is with respect to the parameter s. Theorem 1.2 is an extension of Theorem 3.1 of [3] to polynomials satisfying an m-term recurrence relation instead of a specific four-term recurrence relation as in [3]. The analogous result for orthogonal polynomials satisfying a three-term recurrence is from [17].…”

Section: Polynomials Satisfying An M-term Recurrence Relationmentioning

confidence: 93%

Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

Kuijlaars

Román

2010

Journal of Approximation Theory

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In this paper we consider the model of n non-intersecting squared Bessel processes with parameter α, in the confluent case where all particles start, at time t = 0, at the same positive value x = a, remain positive, and end, at time T = t, at the position x = 0. The positions of the paths have a limiting mean density as n → ∞ which is characterized by a vector equilibrium problem. We show how to obtain this equilibrium problem from different considerations involving the recurrence relations for multiple orthogonal polynomials associated with the modified Bessel functions.We also extend the situation by rescaling the parameter α, letting it increase proportionally to n as n increases. In this case we also analyze the recurrence relation and obtain a vector equilibrium problem for it.

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“…, P −m+2 ≡ 0 and the recurrence coefficients have the scaling limits For the simplest case b ( j) k,n = b ( j) (s), i.e., where we remove the dependence on the parameter n and the recurrence coefficients in (1.5) are actually constant, the zeros of P k = P k,n are closely related to the spectrum of a certain banded Toeplitz matrix. Indeed, if we associate with the functions b ( j) a family of functions 6) and the sequence of k × k Toeplitz matrices {T k (A s )}, k = 0, 1, . .…”

Section: Introductionmentioning

confidence: 99%

The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

Zhang

Román

2011

Journal of Approximation Theory

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We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). On the basis of the four-term recurrence relation, it is shown that, after proper scaling, the sequence of normalized zero counting measures converges weakly to the first component of a vector of two measures which satisfies a vector equilibrium problem with two external fields. We also give the explicit formula for the equilibrium vector in terms of solutions of an algebraic equation.

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Asymptotic zero distribution for a class of multiple orthogonal polynomials

Cited by 43 publications

References 30 publications

An Equilibrium Problem for the Limiting Eigenvalue Distribution of Banded Toeplitz Matrices

An Equilibrium Problem for the Limiting Eigenvalue Distribution of Banded Toeplitz Matrices

Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

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