Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to ${\rm e}^{z}$

Varga, Richard S.; Carpenter, Amos J.

doi:10.1007/s002110050055

Cited by 6 publications

(4 citation statements)

References 11 publications

(22 reference statements)

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“…It shows their remarkable distribution, especially the zeros of q n,n,n distribute themselves in a very particular way. These results on the distribution of the zeros of Hermite-Padé approximants to exponentials may be seen as a continuation of the study initiated by Szegő in [38] concerning the distribution of the zeros of Taylor sections of the series for e z , subsequently generalized by Saff and Varga in [31,32,33] to the zeros of the Padé approximants to e z , see also [22,40].…”

Section: Introductionsupporting

confidence: 54%

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

2004

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We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Padé approximation to the exponential function, defined by p(z)e −z + q(z) + r(z)e z = O(z 3n+2 ) as z → 0. These polynomials are characterized by a RiemannHilbert problem for a 3 × 3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl.

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Section: Introductionsupporting

confidence: 54%

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

2004

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“…Note that contrary to the hypothesis made in [18], no symmetry assumption is made on the set of interpolation points. In connection with the problem of the limiting distribution of zeros, the present results may also be seen as closely related to the study initiated by Szegő in [42] concerning the distribution of the zeros of Taylor sections of the series for e z , subsequently generalized by to the zeros of the Padé approximants to e z (see also [16,44]), and more recently by Stahl in [39,40] to the zeros of the quadratic Hermite-Padé approximants to e z .…”

Section: Introductionmentioning

confidence: 76%

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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“…The behavior of Padé approximants to the exponential function has been studied, among others, in [14,15,16,19,9], and for extensions to Hermite-Padé approximants, one may consult [17,18,12,11]. Generalizations to rational interpolants are investigated in [3,4,2,20,21].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…admits the jumps shown in Figure 3 and has the asymptotic behavior 19) with N the constant matrix defined by (4.6).…”

Section: Airy Model Rh Problemmentioning

confidence: 98%

On sequences of rational interpolants of the exponential function with unbounded interpolation points

Claeys¹,

Wielonsky²

2013

Journal of Approximation Theory

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We consider sequences of rational interpolants r n (z) of degree n to the exponential function e z associated to a triangular scheme of complex points {zj=0 , n > 0, such that, for all n, |z (2n) j | ≤ cn 1−α , j = 0, . . . , 2n, with 0 < α ≤ 1 and c > 0. We prove the local uniform convergence of r n (z) to e z in the complex plane, as n tends to infinity, and show that the limit distributions of the conveniently scaled zeros and poles of r n are identical to the corresponding distributions of the classical Padé approximants. This extends previous results obtained in the case of bounded (or growing like log n) interpolation points. To derive our results, we use the Deift-Zhou steepest descent method for Riemann-Hilbert problems. For interpolation points of order n, satisfying |z (2n) j | ≤ cn, c > 0, the above results are false if c is large, e.g. c ≥ 2π. In this connection, we display numerical experiments showing how the distributions of zeros and poles of the interpolants may be modified when considering different configurations of interpolation points with modulus of order n.

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Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to ${\rm e}^{z}$

Cited by 6 publications

References 11 publications

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

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

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

On sequences of rational interpolants of the exponential function with unbounded interpolation points

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