Zero distributions via orthogonality

Baratchart, Laurent; Küstner, Reinhold; Тотик, Вилмос

doi:10.5802/aif.2130

Cited by 13 publications

(32 citation statements)

References 14 publications

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“…Это обобщает теорему 5.1 и следствие 6.2 из [23]. Основное отличие здесь заключается в добавлении распределения вида (2.22) к мере (2.27), так что у F могут быть полюсы внутри единичного круга.…”

Section: мероморфные приближения комплексных преобразований коши с поunclassified

“…(b) При ξ / ∈ I это утверждение покрывается доказательством леммы 5.2 из [23]. В прочих случаях можно убедится непосредственно, что (3.2) сводится к неравенству 0 π для ξ = b и к π π для ξ ∈ [a, b).…”

Section: мероморфные приближения комплексных преобразований коши с поunclassified

“…(c) Заметим, что ϕ(x) = ψ(x) + Arg(Q(x)) -вещественнозначная функция ограниченной вариации на I, поэтому по лемме 3.4 из [23] существуют много-член T * ̸ = 0 и константа β * ∈ (0, π/16) такие, что…”

Section: мероморфные приближения комплексных преобразований коши с поunclassified

“…(2). С другой стороны, неравенства (3.13) и (3.14) 23) где мы используем равномерную ограниченность {|ω n |} сверху. Мы можем выбрать δ столь малым, что 2τ − Lδ > 0; тогда нижняя грань в (3.22) оказыва-ется больше оценки в (3.23) для больших n. Однако это невозможно, поскольку в силу (3.5) сумма этих двух интегралов равна нулю.…”

Section: теперь приunclassified

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Мероморфные Приближения Комплексных Преобразований Коши С Полярными Особенностями

Барашарт¹,

Baratchart²,

Ятцелев³

et al. 2009

Матем. сб.

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Section: мероморфные приближения комплексных преобразований коши с поunclassified

Section: теперь приunclassified

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Мероморфные Приближения Комплексных Преобразований Коши С Полярными Особенностями

Барашарт¹,

Baratchart²,

Ятцелев³

et al. 2009

Матем. сб.

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“…When h does have zeros (and even in the Hermitian case if it has non-convex support), spurious poles appear whose number can sometimes be estimated from the nature of the zeroing and the smoothness of arg h [42,51,56,8]. But it is S.P.…”

Section: Introductionmentioning

confidence: 99%

Padé approximants to certain elliptic-type functions

Baratchart¹,

Yattselev²

2013

JAMA

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Abstract. Given non-collinear points a 1 , a 2 , a 3 , there is a unique compact ∆ ⊂ C that has minimal logarithmic capacity among all continua joining a 1 , a 2 , and a 3 . For h be a complexvalued non-vanishing Dini-continuous function on ∆, we considerwhere w(z) := 3 k=0 (z − a k ) and w + the one-sided value according to some orientation of ∆. In this work we present strong asymptotics of diagonal Padé approximants to f h , as n → ∞, and describe the behavior of the spurious pole and the regions of locally uniform convergence from a generic perspective.

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Do Orthogonal Polynomials Dream of Symmetric Curves?

Martínez–Finkelshtein

Rakhmanov

2016

Found Comput Math

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Abstract. The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion.The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation.Thus, the second motivation of this paper is to discuss some "mysterious" configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.

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Zero distributions via orthogonality

Cited by 13 publications

References 14 publications

Мероморфные Приближения Комплексных Преобразований Коши С Полярными Особенностями

Мероморфные Приближения Комплексных Преобразований Коши С Полярными Особенностями

Padé approximants to certain elliptic-type functions

Do Orthogonal Polynomials Dream of Symmetric Curves?

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