Roots of complex polynomials and Weyl-Heisenberg frame sets
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- by Peter G. Casazza and Nigel J. Kalton PDF
- Proc. Amer. Math. Soc. 130 (2002), 2313-2318 Request permission
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.References
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Additional Information
- Peter G. Casazza
- Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
- MR Author ID: 45945
- Email: pete@math.missouri.edu
- Nigel J. Kalton
- Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
- Email: nigel@math.missouri.edu
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1896414