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Roots of complex polynomials and Weyl-Heisenberg frame sets


Authors: Peter G. Casazza and Nigel J. Kalton
Journal: Proc. Amer. Math. Soc. 130 (2002), 2313-2318
MSC (1991): Primary 30C15, 11C08, 42C15, 46C05
DOI: https://doi.org/10.1090/S0002-9939-02-06352-9
Published electronically: January 17, 2002
MathSciNet review: 1896414
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Abstract: A Weyl-Heisenberg frame for $L^{2}(\mathbb R)$ is a frame consisting of modulates $E_{mb}g(t) = e^{2{\pi}imbt}g(t)$ and translates $T_{na}g(t) = g(t-na)$, $m,n\in \mathbb Z$, of a fixed function $g\in L^{2} (\mathbb R)$, for $a,b\in \mathbb R$. A fundamental question is to explicitly represent the families $(g,a,b)$ so that $(E_{mb}T_{na}g)_{m,n\in \mathbb Z}$ is a frame for $L^{2}(\mathbb R)$. We will show an interesting connection between this question and a classical problem of Littlewood in complex function theory. In particular, we show that classifying the characteristic functions ${\chi}_{E}$ for which $(E_{m}T_{n}{\chi}_{E})_{m,n\in \mathbb Z}$is a frame for $L^{2}(\mathbb R)$is equivalent to classifying the integer sets $\{n_{1}<n_{2}<\cdots <n_{k}\}$so that $f(z) = \sum_{j=1}^{k} z^{n_{i}}$ does not have any zeroes on the unit circle in the plane.


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Additional Information

Peter G. Casazza
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: pete@math.missouri.edu

Nigel J. Kalton
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: nigel@math.missouri.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06352-9
Received by editor(s): February 28, 2000
Received by editor(s) in revised form: February 16, 2001
Published electronically: January 17, 2002
Additional Notes: The first author was supported by NSF DMS 9706108 and the second author by NSF DMS 9870027
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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