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

DOI:
https://doi.org/10.1090/S0002-9947-07-04186-4

Published electronically:
March 20, 2007

MathSciNet review:
2309193

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Abstract | References | Similar Articles | Additional Information

Abstract: We are concerned with interpolation problems in where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence in the unit disk, we ask whether there exists a nontrivial minorant (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem has a nonvanishing solution whenever for all . The sequences 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 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:
https://doi.org/10.1090/S0002-9947-07-04186-4

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