Interlacing properties of zeros of multiple orthogonal polynomials

Haneczok, Maciej; Assche, Walter Van

doi:10.1016/j.jmaa.2011.11.077

Cited by 30 publications

(29 citation statements)

References 11 publications

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“…Let e i = (0, · · · , 0, 1, 0, · · · , 0) denote the i-th standard unit vector in r dimensions with 1 in the i-th entry. The multiple Meixner polynomials of the first kind obey [5] r recurrence relations of nearest-neighbor form that read as follows:…”

Section: Multiple Meixner Polynomials Of the First Kindmentioning

confidence: 99%

An algebraic model for the multiple Meixner polynomials of the first kind

Miki

Tsujimoto

Vinet

et al. 2012

J. Phys. A: Math. Theor.

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Section: Multiple Meixner Polynomials Of the First Kindmentioning

confidence: 99%

An algebraic model for the multiple Meixner polynomials of the first kind

Miki

Tsujimoto

Vinet

et al. 2012

J. Phys. A: Math. Theor.

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“…The next lemma follows from the proof of theorem 5 in [20] (see also [27]). Proof of theorem 4.1: condition (C).…”

Section: 7mentioning

confidence: 93%

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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“…The interlacing property for the zeros of polynomials orthogonal with respect to a Markov system proved by Kershaw with respect to Lebesgue measure in [47], and under a weak condition, with respect to the Borel measure in [38]. The same argument used in [43] to prove the interlacing property for the type II multiple OPS with respect to measures that form an AT system. Recall that a system of measures (µ 1 , ..., µ r ) forms an AT system for the set of integers (n 1 , ..., n r ) on [a, b] if the measures µ j are absolutely continuous with respect to a measure µ on [a, b], with dµ j (x) = ω j (x)dµ(x) and ω 1 , xω 1 , ..., x n 1 −1 ω 1 , ω 2 , ..., x nr−1 ω r is a Chebyshev system on [a, b] of order n = n 1 + ... + n r − 1 [48,60].…”

Section: Some Properties Of Zerosmentioning

confidence: 98%

“…We are interested in the zeros of OPS because of their important role mainly in interpolation and approximation theory, Gauss-Jacobi quadrature, spectral theory, and in image analysis and pattern recognition. The first real investigation of the zeros of multiple OPS was not performed till 2011, when Haneczok with Van Assche gave sufficient conditions for zeros of the latter class to be real and distinct [43], a result we shall build upon and generalize to the d-orthogonality with the aid of oscillation matrices which involves further, some interlacing properties.…”

Section: Introductionmentioning

confidence: 99%

Some Inverse Problems for d-Orthogonal Polynomials

Saib

Zerouki

2012

Mediterr. J. Math.

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The aim of this paper are two folds. The first part is concerned with the associated and the so-called co-polynomials, i.e., new sequences obtained when finite perturbations of the recurrence coefficients are considered. Moreover, the second part deals with Darboux factorization of Jacobi matrices. Here the respective co-polynomials solutions are explicitly expressed in terms of the fundamental solutions of a (d+2)-term recurrence relation. New identities and formulas related to determinants with co-polynomials entries are obtained. Accordingly, further determinants bring out partial generalizations of Christoffel Darboux formula. Some of new sequences proved useful for determining the entries of matrices in LU and UL decomposition of Jacobi matrix. The last one gives rise of a d-analogue of kernel polynomials with quite a few properties, and further a new characterization of the d-quasiorthogonality. Kernel polynomials also appear in the (d+1)-decomposition of a d-symmetric sequence. Exploiting properties of d-symmetric sequences, reveal a simple proof of Darboux factorizations. It terms out that Jacobi matrix for d-OPS is a product of d lower bidiagonal matrices and one upper bidiagonal matrix and that each lower bidiagonal matrix is in fact a closed connection between two adjacent components for some d-(symmetric)OPS. Furthermore, we pointed out that if the first component is Hahn classical d-OPS then the corresponding d-symmetric sequence as well as all the components are Hahn classical d-OPS as well. Oscillation matrices assert that zeros of d-OPS are positive and simple whenever the recurrence coefficients are strict positive. Further interlacing properties are justified by the same approach.2010 Mathematics Subject Classification. Primary 42C05; Secondary 33C45.

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Interlacing properties of zeros of multiple orthogonal polynomials

Cited by 30 publications

References 11 publications

An algebraic model for the multiple Meixner polynomials of the first kind

An algebraic model for the multiple Meixner polynomials of the first kind

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

Some Inverse Problems for d-Orthogonal Polynomials

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