Agler, J, Lykova, ZA and Young, NJ (2015) 3-extremal holomorphic maps and the symmetrised bidisc. Journal of Geometric Analysis, 25 (3). pp. 2060-2102. ISSN 1050-6926
Abstract
We analyze the 3-extremal holomorphic maps from the unit disc D to the symmetrized bidisc G=def{(z+w,zw):z,w∈D} with a view to the complex geometry and function theory of G. These are the maps whose restriction to any triple of distinct points in D yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational G-inner functions of degree at most 4, to be called aligned and caddywhompus functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are 3-extremal. We describe a method for the construction of aligned rational G-inner functions; with the aid of this method we reduce the solution of a 3-point interpolation problem for aligned holomorphic maps from D to G to a collection of classical Nevanlinna–Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for G.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2014, Mathematica Josephina, Inc. This is an author produced version of a paper published in Journal of Geometric Analysis. The final publication is available at Springer via http://dx.doi.org/10.1007/s12220-014-9504-3. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Extremal holomorphic maps; Symmetrised bidis; G-inner functions; Holomorphic interpolation; Invariant distances; μ-synthesis |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number EPSRC EP/J004545/1 Newcastle University/EPSRC BH 122321 |
Depositing User: | Symplectic Publications |
Date Deposited: | 17 Feb 2016 13:43 |
Last Modified: | 31 Jan 2018 19:41 |
Published Version: | http://dx.doi.org/10.1007/s12220-014-9504-3 |
Status: | Published |
Publisher: | Springer Verlag |
Identification Number: | 10.1007/s12220-014-9504-3 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:94776 |