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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Inner functions and zero sets for $ \ell^{p}_{A}$


Authors: Raymond Cheng, Javad Mashreghi and William T. Ross
Journal: Trans. Amer. Math. Soc. 372 (2019), 2045-2072
MSC (2010): Primary 30B10, 30C75, 30H10, 30J05
DOI: https://doi.org/10.1090/tran/7675
Published electronically: April 25, 2019
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Abstract: In this paper we characterize the zero sets of functions from $ \ell ^{p}_{A}$ (the analytic functions on the open unit disk $ \mathbb{D}$ whose Taylor coefficients form an $ \ell ^p$ sequence) by developing a concept of an ``inner function'' modeled by Beurling's discussion of the Hilbert space $ \ell ^{2}_{A}$, the classical Hardy space. The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, we give an alternative proof of a result of Vinogradov [Dokl. Akad. Nauk SSSR 160 (1965), pp. 263-266] which says that when $ p > 2$, there are zero sets for $ \ell ^{p}_{A}$ which are not Blaschke sequences.


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Additional Information

Raymond Cheng
Affiliation: Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529
Email: rcheng@odu.edu

Javad Mashreghi
Affiliation: Département de mathématiques et de statistique, Université laval, Québec, Canada, G1V 0A6
Email: javad.mashreghi@mat.ulaval.ca

William T. Ross
Affiliation: Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia 23173
Email: wross@richmond.edu

DOI: https://doi.org/10.1090/tran/7675
Received by editor(s): February 13, 2018
Received by editor(s) in revised form: July 24, 2018, and July 26, 2018
Published electronically: April 25, 2019
Additional Notes: This work was supported by NSERC (Canada).
Article copyright: © Copyright 2019 American Mathematical Society