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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Regions of meromorphy determined by the degree of best rational approximation
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by E. B. Saff PDF
Proc. Amer. Math. Soc. 29 (1971), 30-38 Request permission

Abstract:

In this paper we investigate the relationship between the degree of best rational approximation to a given function $f(z)$ and the regions in which $f(z)$ is meromorphic. We show, for example, that if rational functions ${r_{nv}}(z)$ of respective types (n, v), i.e., rational functions with v free poles, converge geometrically (as $n \to \infty$) to $f(z)$ on a closed Jordan region E, then $f(z)$ must be meromorphic in a region which contains E in its interior.
References
    J. Hadamard, “Essai sur l’étude des fonctions données par leur développement de Taylor,” in Oeuvres de Jacques Hadamard. Vol. 1, Centre National de la Recherche Scientifique, Paris, 1968. MR 37 #6158. A. I. Markuševič, Theory of functions of a complex variable. Vol. 3, GITTL, Moscow, 1950; English transl., Prentice-Hall, Englewood Cliffs, N. J., 1967. MR 12, 87; MR 35 #6799.
  • J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587
  • J. L. Walsh, The convergence of sequences of rational functions of best approximation with some free poles, Approximation of Functions (Proc. Sympos. General Motors Res. Lab., 1964) Elsevier Publ. Co., Amsterdam, 1965, pp. 1–16. MR 0186986
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 29 (1971), 30-38
  • MSC: Primary 30.70
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0281930-2
  • MathSciNet review: 0281930