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Extremely weak interpolation in $ H^\infty$


Author: Andreas Hartmann
Journal: Proc. Amer. Math. Soc. 140 (2012), 2411-2416
MSC (2010): Primary 30E05, 32A35
Published electronically: April 20, 2011
MathSciNet review: 2898703
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Abstract: Given a sequence of points in the unit disk, a well known result due to Carleson states that if given any point of the sequence it is possible to interpolate the value one in that point and zero in all the other points of the sequence, with uniform control of the norm in the Hardy space of bounded analytic functions on the disk, then the sequence is an interpolating sequence (i.e. every bounded sequence of values can be interpolated by functions in the Hardy space). It turns out that such a result holds in other spaces. In this short paper we would like to show that for a given sequence it is sufficient to find just one function suitably interpolating zeros as well as ones to deduce interpolation in the Hardy space. The result has an interesting interpretation in the context of model spaces.


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Additional Information

Andreas Hartmann
Affiliation: Equipe d’Analyse, Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France
Email: hartmann@math.u-bordeaux.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10851-7
Keywords: Hardy spaces, interpolating sequences, weak interpolation
Received by editor(s): October 16, 2010
Received by editor(s) in revised form: October 18, 2010, and February 22, 2011
Published electronically: April 20, 2011
Additional Notes: This project was elaborated while the author was Gaines Visiting Chair at the University of Richmond and partially supported by the French ANR-project FRAB
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.