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

Abstract

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.

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

“…This allows us to obtain uniform asymptotics for the polynomials P n and Q n , from which asymptotics for the original polynomials p n and q n follow. Such asymptotics were obtained in [21] for interpolation points satisfying (1.7), and can be obtained similarly in our more general situation.…”

“…In particular, if ρ n = (n) √ n with (n) which tends to 0 as n tends to infinity, then (1.10) can be rewritten as e z + r n (z) = (−1) n e 4n 2n+1 w 2n+1 (z)e z−1 1 + Ø( 2 (n)) + Ø 1 n α . (1.11) Remark 1.2 For the special case of bounded interpolation points, the error estimate (1.11) agrees with the estimate (2.4) of [21] except for a minus sign that was incorrect there.…”

“…, 2n}, n ∈ N. (1.6) If the interpolation points do not grow too rapidly with n, i.e. if there exist constants 0 < α ≤ 1 and c > 0 such that 7) it was proved in [21,Theorem 2.2] that a pair (p n , q n ), such that (1.4)-(1.5) hold true (with n 1 = n 2 = n), satisfies p n (z) → −e z/2 , q n (z) → e −z/2 , locally uniformly in C, where q n is normalized so that q n (0) = 1. In particular, p n (z)/q n (z) converges to −e z uniformly on compact sets in the complex plane as n → ∞.…”

Type II Hermite–Padé approximation to the exponential function

“…It was followed by the papers [1,3] which dealt the asymptotic analysis of 3 × 3 matrix valued Riemann-Hilbert problems arising in random matrix theory. A Riemann-Hilbert analysis for rational interpolants for the exponential function was carried out in [24]. It is the aim of this paper to show that the Riemann-Hilbert analysis of [9] also produces the corresponding asymptotic results for the scaled type II Hermite-Padé polynomials A n (z) = a n,n,n (3nz), B n (z) = b n,n,n (3nz), C n (z) = c n,n,n (3nz), (1.4) and for the type II remainder terms E (1) n (z) = A n (z)e −3nz − B n (z), E (2) n (z) = A n (z)e 3nz − C n (z).…”

Аппроксимации Паде Функций Миттаг-Леффлера

“…Подробный обзор этих резуль-татов имеется и в цитируемых выше статьях А. А. Гончара, Е. А. Рахманова [19] и А. И. Аптекарева [20] (см. также работы [24]- [26]). Перейдем теперь непосредственно к формулировкам основных результатов настоящей работы.…”