Free interpolation by nonvanishing analytic functions
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- by Konstantin Dyakonov and Artur Nicolau PDF
- Trans. Amer. Math. Soc. 359 (2007), 4449-4465 Request permission
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 $1\ge |a_j|\ge \varepsilon _j$ 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 $H^\infty \cap \operatorname {VMO}$.References
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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
- 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.
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2309193