Email this article A direct proof of Stahl's theorem for a generic class of algebraic functions
- Authors: Suetin S.P.1
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 213, No 11 (2022)
- Pages: 102-117
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133495
- DOI: https://doi.org/10.4213/sm9649
- ID: 133495
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Abstract
Under the assumption that Stahl's $S$-compact set exists we give a short proof of the limiting distribution of the zeros of Pade polynomials and the convergence in capacity of diagonal Pade approximants for a generic class of algebraic functions. The proof is direct, rather than by contradiction as Stahl's original proof was. The ‘generic class’ means, in particular, that all the ramification points of the multisheeted Riemann surface of the algebraic function in question are of the second order (that is, all branch points of the function are of square root type). As a consequence, a conjecture of Gonchar relating to Pade approximations is proved for this class of algebraic functions. We do not use the relations of orthogonality for Pade polynomials. The proof is based on the maximum principle only. Bibliography: 19 titles.
About the authors
Sergey Pavlovich Suetin
Steklov Mathematical Institute of Russian Academy of Sciences
Email: suetin@mi-ras.ru
Doctor of physico-mathematical sciences, no status
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