Pade approximants for functions with branch points - strong asymptotics of Nuttall-Stahl polynomials

• 成果类型: Article
• 署名作者: Aptekarev, Alexander I.; Yattselev, Maxim L.
• 署名单位: Russian Academy of Sciences; Keldysh Institute of Applied Mathematics; Purdue University System; Purdue University; Purdue University in Indianapolis
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-016-0133-5
• 发表日期: 2015
• 页码: 217-280
• 关键词: riemann-hilbert approach; Orthogonal polynomials; CONVERGENCE; Respect

摘要

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, , . 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 Pad, approximants for f. The Pad, 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 . 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 Pad, 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.