Acta Mathematica

Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials

Alexander I. Aptekarev and Maxim L. Yattselev

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Abstract

Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, fA(C¯\A), #A<. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Padé approximants for f. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function fA(C¯\A). The complete proof of Nuttall’s conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro-logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

Article information

Source
Acta Math., Volume 215, Number 2 (2015), 217-280.

Dates
Received: 16 July 2012
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485802454

Digital Object Identifier
doi:10.1007/s11511-016-0133-5

Mathematical Reviews number (MathSciNet)
MR3455234

Zentralblatt MATH identifier
0863.94012

Subjects
Primary: 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
Secondary: 41A20: Approximation by rational functions 41A21: Padé approximation

Keywords
Padé approximation orthogonal polynomials non-Hermitian orthogonality strong asymptotics

Rights
2016 © Institut Mittag-Leffler

Citation

Aptekarev, Alexander I.; Yattselev, Maxim L. Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials. Acta Math. 215 (2015), no. 2, 217--280. doi:10.1007/s11511-016-0133-5. https://projecteuclid.org/euclid.acta/1485802454


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