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.
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References
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© 1984 Springer-Verlag
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Iserles, A. (1984). Order stars, contractivity and a pick-type theorem. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072404
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DOI: https://doi.org/10.1007/BFb0072404
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