Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

Kuijlaars, Arno B. J.; Assche, Walter Van; Wielonsky, Franck

doi:10.1007/s00365-004-0579-0

Cited by 39 publications

(78 citation statements)

References 45 publications

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“…As in the case of orthogonal polynomials [33,34], the local parametrix will then be constructed with the help of Airy functions. Also, for larger-size RH problems, Airy parametrices have been constructed; see, e.g., [3,19,28,59]. The situation in the present case is similar, and so we will not give all details of the construction here.…”

Section: Construction Of Local Parametricesmentioning

confidence: 93%

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Duits

Kuijlaars

2008

Comm Pure Appl Math

133

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The eigenvalue statistics of a pair .M 1 ; M 2 / of n n Hermitian matrices taken randomly with respect to the measurecan be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 4 matrix-valued RiemannHilbert problem characterizing one of the families of biorthogonal polynomials in the special case W .y/ D y 4 =4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M 1 (when averaged over M 2 ) in the global and local regime as n ! 1 in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.

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Section: Construction Of Local Parametricesmentioning

confidence: 93%

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Duits

Kuijlaars

2008

Comm Pure Appl Math

133

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“…After the present paper had been submitted for publication, another approach to the analysis of strong asymptotics of quadratic Hermite-Pade´polynomials of type I has been undertaken in a paper by Kuijlaars et al, [11], which is based on a Riemann-Hilbert problem. An announcement of this result has already been published in [12].…”

Section: Article In Pressmentioning

confidence: 98%

Quadratic Hermite–Padé polynomials associated with the exponential function

Stahl

2003

Journal of Approximation Theory

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The asymptotic behavior of quadratic Hermite-Pade´polynomials p n ; q n ; r n AP n associated with the exponential function is studied for n-N: These polynomials are defined by the relation p n ðzÞ þ q n ðzÞe z þ r n ðzÞe 2z ¼ Oðz 3nþ2 Þ as z-0; ðÃÞwhere OðÁÞ denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in ðÃÞ; and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite-Pade´approximants, which will be done in a follow-up paper. r

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“…We follow an approach similar to the one used in [27] for studying quadratic Hermite-Padé approximants to e z . The asymptotic analysis is based on a Riemann-Hilbert formulation for the polynomials P n and Q n , combined with a steepest descent analysis for Riemann-Hilbert problems.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function

Wielonsky

2004

Journal of Approximation Theory

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We study the asymptotic behavior of the polynomials p and q of degrees n, rational interpolants to the exponential function, defined by p(z)e −z/2 + q(z)e z/2 = O ( 2n+1 (z)), as z tends to the roots of 2n+1 , a complex polynomial of degree 2n + 1. The roots of 2n+1 may grow to infinity with n, but their modulus should remain uniformly bounded by c log(n), c < 1/2, as n → ∞. We follow an approach similar to the one in a recent work with Arno Kuijlaars and Walter Van Assche on Hermite-Padé approximants to e z . The polynomials p and q are characterized by a Riemann-Hilbert problem for a 2×2 matrix valued function. The Deift-Zhou steepest descent method for Riemann-Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials p(2nz) and q(2nz) in every domain in the complex plane. From these asymptotics, we deduce uniform convergence of general rational interpolants to the exponential function and a precise estimate on the error function. This extends previous results on rational interpolants to the exponential function known so far for real interpolation points and some cases of complex conjugate interpolation points.

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Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

Cited by 39 publications

References 45 publications

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Quadratic Hermite–Padé polynomials associated with the exponential function

Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function

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