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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Free interpolation by nonvanishing analytic functions


Authors: Konstantin Dyakonov and Artur Nicolau
Journal: Trans. Amer. Math. Soc. 359 (2007), 4449-4465
MSC (2000): Primary 46J15, 30D50, 30H05
Published electronically: March 20, 2007
MathSciNet review: 2309193
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Abstract: We are concerned with interpolation problems in $ H^\infty$ where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence $ \{z_j\}$ in the unit disk, we ask whether there exists a nontrivial minorant $ \{\varepsilon_j\}$ (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem $ f(z_j)=a_j$ has a nonvanishing solution $ f\in H^\infty$ whenever $ 1\ge\vert a_j\vert\ge\varepsilon_j$ for all $ j$. The sequences $ \{z_j\}$ with this property are completely characterized. Namely, we identify them as `` thin" sequences, a class that arose earlier in Wolff's work on free interpolation in $ H^\infty\cap$VMO.


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Additional Information

Konstantin Dyakonov
Affiliation: ICREA and Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, E-08007 Barcelona, Spain
Email: dyakonov@mat.ub.es

Artur Nicolau
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain
Email: artur@mat.uab.es

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04186-4
PII: S 0002-9947(07)04186-4
Keywords: Nonvanishing analytic functions, thin interpolating sequences
Received by editor(s): October 11, 2004
Received by editor(s) in revised form: October 1, 2005
Published electronically: March 20, 2007
Additional Notes: Both authors were supported by the European Community’s Human Potential Program under contract HPRN-CT-2000-00116 (Analysis and Operators). The first author was also supported by DGICYT Grant MTM2005-08984-C02-02, CIRIT Grant 2005-SGR-00611, Grant 02-01-00267 from the Russian Foundation for Fundamental Research, and by the Ramón y Cajal program (Spain). The second author was supported by DGICYT Grant MTM2005-00544 and CIRIT Grant 2005-SGR-00774.
Article copyright: © Copyright 2007 American Mathematical Society