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Order stars, contractivity and a pick-type theorem

  • Arieh Iserles
Approximation And Interpolation Theory
  • 482 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1105)

Abstract

Given a function f that is analytic in the complex domain V and such that |f|≡1 along ∂V (with the possible exception of essential singularities) we examine analytic approximations R to f that are contractions in cℓV. By applying the theory of order stars we demonstrate that the nature of essential singularities and zeros of f imposes surprisingly severe upper bounds on the degree of interpolation by a contractive approximation R. It is proved that, subject to V being conformal to the unit disk, contractive interpolations that satisfy the given bounds are attained by rational functions. Finally, we apply our theory to prove a version of the classical Pick theorem that is valid in every complex domain.

Keywords

Unit Disk Complex Domain Interpolation Point Straightforward Consequence Analytic Boundary 
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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Arieh Iserles
    • 1
  1. 1.King's CollegeUniversity of CambridgeCambridgeEngland

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