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Runge Theory for Compact Sets

  • Reinhold Remmert
Part of the Graduate Texts in Mathematics book series (GTM, volume 172)

Abstract

In discs B, all holomorphic functions are approximated compactly by their Taylor polynomials. In particular, for every fO(B) and every compact set K in B, there exists a sequence of polynomials pnsuch that lim |f - Pn|K = 0. In arbitrary domains, polynomial approximation is not always possible; in C×, for example, there is no sequence of polynomials pn that approximates the holomorphic function l/z uniformly on a circle γ, for it would then follow that
$$$$2\pi i = \int_\gamma{\frac{{d\varsigma }}{\varsigma }}= \lim \int_\gamma{{p_n}\left( \varsigma\right)d\varsigma }= 0$$

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Reinhold Remmert
    • 1
  1. 1.Mathematisches InstitutWestfälische Wilhelms—Universität MünsterMünsterGermany

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