On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials

Aptekarev, A. I.; Lagomasino, G. López; Martínez–Finkelshtein, Andrei

doi:10.1070/rm9769

Cited by 14 publications

(12 citation statements)

References 39 publications

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“…In the asymptotic analysis of multiple orthogonal polynomials with respect to a general Nikishin system N (σ 1 , σ 2 ) in which σ 2 is discrete, the associated model vector equilibrium problem exhibits an external field acting on supp(σ 1 ) plus a constraint on σ 2 . This situation is encountered, for example, in [30], as well as for Pollaczek weights w j in [40]; see [4] for the analysis of a general case. Let µ be a positive Borel measure with support contained in R and satisfying…”

Section: Riemann-hilbert Characterizationmentioning

confidence: 99%

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

Aptekarev

Lagomasino

Martínez–Finkelshtein

2015

Dokl. Math.

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Abstract-The asymptotic properties of multiple orthogonal polynomials with respect to two Pollaczek weights with different parameters are considered. This set of weights is a Nikishin system generated by two measures with unbounded supports; moreover, the second measure is discrete. During the last years, multiple orthogonal polynomials with respect to Nikishin systems of this type have found wide applications in the the ory of random matrices. Strong asymptotic formulas for the polynomials under consideration are obtained by means of the matrix Riemann-Hilbert method.

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Section: Riemann-hilbert Characterizationmentioning

confidence: 99%

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

Aptekarev

Lagomasino

Martínez–Finkelshtein

2015

Dokl. Math.

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“…Finally we notice that the asymptotic properties of Hermite-Padé polynomials are still of unabated interest (see [1], [5], [22], [34], [23], [34], [4] and the bibliography therein). An accessible presentation of the potential theory on Riemann surface is given in the papers [9]- [11].…”

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confidence: 99%

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

Suetin

2021

Preprint

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In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987Stahl's novel approach, -1988, to explore Hermite-Padé polynomials based on Riemann surface properties.In particular, we explore the limit zero distribution of type I Hermite-Padé polynomials Qn,0, Qn,1, Qn,2, deg Qn,j n, for a collection of three analytic elements [1, f∞, f 2 ∞ ]. The element f∞ is an element of a function f from the class C(z, w) where w is supposed to be from the class Z ±1/2 ([−1, 1]) of multivalued analytic functions generated by the inverse Zhukovskii function with the exponents from the set {±1/2}. The Riemann surface corresponding to f ∈ C(z, w) is a four-sheeted Riemann surface R4(w) and all branch points of f are of the first order (i.e., all branch points are of square root type).It is proved that there exits the similar limit zero distribution for all three polynomials Qn,j. The answer is done in terms of Nuttall's condenser which was introduced by E. Rakhmanov and the author in 2013. The corresponding limit measure is supported on the second plate of Nuttall's condenser which coincides with the projection of the boundary between the first and the second sheets of the three-sheeted Riemann surface R3(w∞) associated by Nuttall with the element w∞. The limit measure is a unique solution of a mixed Green-logarithmic theoretical-potential equilibrium problem. In general the surface R3(w∞) can be of any genus.Since the algebraic function f ∈ C(z, w) is of fourth order and we consider the triple of the analytic elements [1, f∞, f 2 ∞ ] but not the quadruple [1, f∞, f 2 ∞ , f 3 ∞ ] ones, the result is new and does not follow from the known results.As in previous paper arXiv: 2108.00339 and following to Stahl's ideas, 1987-1988, we do not use the orthogonality relations at all. The proof is based on the maximum principle only.Bibliography: [45] titles.

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“…For more information about multiple orthogonal polynomials, we refer the reader to survey papers [45,15,6] and monograph [44]. For some recent results in the theory of MOPs, we refer the reader to [41,48,21,33,53,34,37,38,5,32,47].…”

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confidence: 99%

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

Aptekarev

Denisov

Yattselev

2019

Trans. Amer. Math. Soc.

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We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on Z + to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory. Contents arXiv:1806.10531v1 [math.CA] 27 Jun 2018where Y (p) has d children each corresponding to the index i ∈ {1, . . . , d}.First, we claim that J κ, N → J κ, c in the strong operator sense, i.e.,for every fixed f ∈ 2 (V). Indeed, let χ |X|<ρ be the characteristic function of the ball in T with center at O and radius ρ. Given any > 0, there is ρ such thatSince coefficients {a n,j } and {b n,j } are uniformly bounded, we haveuniformly in N . Having and ρ fixed, we getby our assumptions (4.19). This proves our claim. Next, the Second Resolvent Identity from perturbation theory of operators givesfor z ∈ C\R. Since (J ej , c − z) −1 | Im z| −1 by the Spectral Theorem, we can take | N | → ∞ and use the above claim to obtain (4.21) lim

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On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials

Cited by 14 publications

References 39 publications

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

Strong asymptotics for the Pollaczek multiple orthogonal polynomials

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

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