Zero Distribution of Hermite-Padé Polynomials and Convergence Properties of Hermite Approximants for Multivalued Analytic Functions

Ikonomov, Nikolay; Kovacheva, Ralitza; Suetin, Sergey Pavlovich

doi:10.48550/arxiv.1603.03314

Cited by 2 publications

(5 citation statements)

References 42 publications

(87 reference statements)

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“…If λ is an equilibrium measure, then P λ + ψ w F everywhere on S(λ), which shows that λ ∈ M • 1 (F ). Since the sets of zero inner capacity play no role in integration with respect to measures in M • 1 (F ), we obtain (23). Finally, (P λ + ψ)(z) ≡ w F on S(λ), because F is a regular compact set.…”

Section: Proof Of Theoremmentioning

confidence: 92%

“…An appeal to (22) with ε = 1 shows that any measure ν ∈ M • 1 (F ) satisfying ( 23) minimizes the energy integral J ψ (•). If a measure λ satisfies condition (23), then it obeys the equilibrium relations (16) with…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Remark 3. It is worth pointing out that the approach proposed in the present paper stems, to some extent, from the analysis of numerical experiments of [21]- [23].…”

Section: We Now Setmentioning

confidence: 99%

See 2 more Smart Citations

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

Suetin

2018

Proc. Steklov Inst. Math.

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A new approach to the problem of the zero distribution of Hermite-Padé polynomials of type I for a pair of functions f1, f2 forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field, which is posed on a two-sheeted Riemann surface.Bibliography: [50] titles. Paper: http://mi.mathnet.ru/eng/tm3908

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Section: Proof Of Theoremmentioning

confidence: 92%

Section: Proof Of Theoremmentioning

confidence: 99%

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On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

Suetin

2018

Proc. Steklov Inst. Math.

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“…In such settings, the mere existence of solution of the corresponding vector max-min problems is not guaranteed. Hence, despite many efforts and recent contributions [3,6,9,26,27,29,33,41,43], it looks like a multidimensional extension of the GRS theory, valid for MOPs with arbitrary number of weights and no underlying symmetries, is nowhere in sight. This is why the analysis of even simplest non-trivial cases that do not exhibit any particular symmetry can shed a new light onto the problem.…”

Section: Introductionmentioning

confidence: 99%

“…It should be pointed out that the large degree zero distribution of these polynomials is highly non-trivial, exhibiting several phase transitions, as Figures 1 and 2 illustrate. Some previous contributions [3,9] addressed the large degree asymptotics on the plane in the non-symmetric case and for weights on bounded sets having only finite branch points; see also [26,27] for nice numerical experiments and empirical discussion related to the so-called Nutall's conjecture. To our knowledge, the present work is the first systematic study of zero distribution of MOPs with complex zeros that exhibit non-hermitian orthogonality on unbounded sets (leading to consideration of extremal problems with an external field) with no real symmetry.…”

Section: Introductionmentioning

confidence: 99%

Critical measures for vector energy: Asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight

Martínez–Finkelshtein

Silva

2019

Advances in Mathematics

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We consider the type I multiple orthogonal polynomials (MOPs) (An,m, Bn,m), deg An,m ≤ n − 1, deg Bn,m ≤ m − 1, and type II MOPs Pn,m, deg Pn,m = n + m, satisfying non-hermitian orthogonality with respect to the weight e −z 3 on two unbounded contours γ 1 and γ 2 on C, with (in the case of type II MOPs) n conditions on γ 1 and m on γ 2 . Under the assumption that n, m → ∞, n n + m → α ∈ (0, 1)we find the detailed (rescaled) asymptotics of An,m, Bn,m and Pn,m on C, and describe the phase transitions of this limit behavior as a function of α. This description is given in terms of the vector critical measure µα = (µ 1 , µ 2 , µ 3 ), the saddle point of an energy functional comprising both attracting and repelling forces. This critical measure is characterized by a cubic equation (spectral curve), and its components µ j live on trajectories of a canonical quadratic differential on the Riemann surface of this equation. The structure of these trajectories and their deformations as functions of α was object of study in our previous paper [Adv. Math. 302 (2016), 1137-1232, and some of the present results strongly rely on the analysis carried out there.We conclude that the asymptotic zero distribution of the polynomials An,m and Pn,m are given by appropriate combinations of the components µ j of the vector critical measure µα. However, in the case of the zeros of Bn,m the behavior is totally different, and can be described in terms of the balayage of the signed measure µ 2 − µ 3 onto certain curves on the plane. These curves are constructed with the aid of the canonical quadratic differential , and their topology has three very distinct characters, depending on the value of α, and are obtained from the critical graph of .Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the Deift-Zhou nonlinear steepest descent analysis of a 3 × 3 matrix-valued Riemann-Hilbert problem characterizing both (An,m, Bn,m) and Pn,m, which allows us to obtain the limit behavior of both types of MOPs simultaneously.We illustrate our findings with results of several numerical experiments, explain the computational methodology, and formulate some conjectures and empirical observations based on these experiments.

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Zero Distribution of Hermite-Padé Polynomials and Convergence Properties of Hermite Approximants for Multivalued Analytic Functions

Cited by 2 publications

References 42 publications

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

Critical measures for vector energy: Asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight

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