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• W2007014797 abstract "We introduce the class of n-extremal holomorphic maps, a class that generalizes both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence 𝒞ν, where ν⩾0, of necessary conditions for solvability, prove that they are of strictly increasing strength and show that 𝒞n−3 is insufficient for the solvability of an n-point problem for n⩾3. We propose the conjecture that condition 𝒞n−2 is necessary and sufficient for the solvability of an n-point interpolation problem for Γ and we explore the implications of this conjecture. We introduce a classification of rational Γ-inner functions, that is, analytic functions from the disc into Γ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of Γ. The classes are related to n-extremality and the conditions 𝒞ν; we prove numerous strict inclusions between the classes." @default.
• W2007014797 created "2016-06-24" @default.
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• W2007014797 date "2012-10-26" @default.
• W2007014797 modified "2023-10-18" @default.
• W2007014797 title "Extremal holomorphic maps and the symmetrized bidisc" @default.
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• W2007014797 doi "https://doi.org/10.1112/plms/pds049" @default.
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