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Transactions of the American Mathematical Society

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Nonnegativity constraints for structured complete systems


Authors: Alexander M. Powell and Anneliese H. Spaeth
Journal: Trans. Amer. Math. Soc. 368 (2016), 5783-5806
MSC (2010): Primary 42C80; Secondary 46E30, 46B15
DOI: https://doi.org/10.1090/tran/6562
Published electronically: December 3, 2015
MathSciNet review: 3458399
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Abstract: We investigate pointwise nonnegativity as an obstruction to various types of structured completeness in $ L^p(\mathbb{R})$. For example, we prove that if each element of the system $ \{f_n\}_{n=1}^\infty \subset L^p(\mathbb{R})$ is pointwise nonnegative, then $ \{f_n\}_{n=1}^{\infty }$ cannot be an unconditional basis or unconditional quasibasis (unconditional Schauder frame) for $ L^p(\mathbb{R})$. In particular, in $ L^2(\mathbb{R})$ this precludes the existence of nonnegative Riesz bases and frames. On the other hand, there exist pointwise nonnegative conditional quasibases in $ L^p(\mathbb{R})$, and there also exist pointwise nonnegative exact systems and Markushevich bases in $ L^p(\mathbb{R})$.


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Additional Information

Alexander M. Powell
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: alexander.m.powell@vanderbilt.edu

Anneliese H. Spaeth
Affiliation: Department of Mathematics, Huntingdon College, Montgomery, Alabama 36106
Email: aspaeth@hawks.huntingdon.edu

DOI: https://doi.org/10.1090/tran/6562
Received by editor(s): November 15, 2013
Received by editor(s) in revised form: July 24, 2014
Published electronically: December 3, 2015
Additional Notes: The authors were supported in part by NSF DMS Grant 1211687.
Article copyright: © Copyright 2015 American Mathematical Society