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Spurious Poles in Diagonal Rational Approximation

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Progress in Approximation Theory

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 19))

Abstract

Any function f meromorphic in C admits fast rational approximation. That is, if K is a compact set in which f is analytic, there exist rational functions R n of type (n,n),n ≥ 1, such that

$$\mathop {\lim }\limits_{n \to \infty } \left\| {f - {R_n}} \right\|_{{L_\infty }(K)}^{1/n} = 0$$

. More generally, any function f defined on an open set U, and admitting such approximation on a compact K ⊂ U with positive logarithmic capacity, is said to belong to the Gonchar-Walsh Class on U. We discuss at an introductory, non-technical, level, the problem of spurious poles for diagonal and sectorial sequences of rational approximants to functions in the Gonchar-Walsh class. In particular, we concentrate on some recent positive results on the distribution of poles, and some of their consequences.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Witwatersrand University, WITS 2050, Johannesburg, Rep. South Africa

    D. S. Lubinsky

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  1. D. S. Lubinsky
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Editors and Affiliations

  1. Steklov Mathematics Institute, 117966, Moscow GSP-1, Russia

    A. A. Gonchar

  2. Institute for Constructive Mathematics, University of South Florida, 33620, Tampa, FL, USA

    E. B. Saff

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© 1992 Springer-Verlag New York, Inc.

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Lubinsky, D.S. (1992). Spurious Poles in Diagonal Rational Approximation. In: Gonchar, A.A., Saff, E.B. (eds) Progress in Approximation Theory. Springer Series in Computational Mathematics, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2966-7_8

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  • DOI: https://doi.org/10.1007/978-1-4612-2966-7_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7737-8

  • Online ISBN: 978-1-4612-2966-7

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