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

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- by Raymond Cheng, Javad Mashreghi and William T. Ross PDF
- Trans. Amer. Math. Soc.
**372**(2019), 2045-2072 Request permission

## 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.## References

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## Additional Information

**Raymond Cheng**- Affiliation: Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529
- MR Author ID: 317015
- Email: rcheng@odu.edu
**Javad Mashreghi**- Affiliation: Département de mathématiques et de statistique, Université laval, Québec, Canada, G1V 0A6
- MR Author ID: 679575
- Email: javad.mashreghi@mat.ulaval.ca
**William T. Ross**- Affiliation: Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia 23173
- MR Author ID: 318145
- Email: wross@richmond.edu
- 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).
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 2045-2072 - MSC (2010): Primary 30B10, 30C75, 30H10, 30J05
- DOI: https://doi.org/10.1090/tran/7675
- MathSciNet review: 3976584