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Roots of complex polynomials and Weyl-Heisenberg frame sets. (English) Zbl 0991.42023

Summary: 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 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.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
11C08 Polynomials in number theory
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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[1] Peter Borwein and Tamás Erdélyi, On the zeros of polynomials with restricted coefficients, Illinois J. Math. 41 (1997), no. 4, 667 – 675. · Zbl 0906.30005
[2] Peter Borwein, Tamás Erdélyi, and Géza Kós, Littlewood-type problems on [0,1], Proc. London Math. Soc. (3) 79 (1999), no. 1, 22 – 46. · Zbl 1039.11046 · doi:10.1112/S0024611599011831
[3] Peter G. Casazza, Ole Christensen, and A. J. E. M. Janssen, Classifying tight Weyl-Heisenberg frames, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999) Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 131 – 148. · Zbl 0960.42006 · doi:10.1090/conm/247/03800
[4] P.G. Casazza and M. Lammers, Classifying characteristic functions giving Weyl-Heisenberg frames, Proceedings SPIE, San Diego (2000).
[5] Hans G. Feichtinger and Thomas Strohmer , Gabor analysis and algorithms, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 1998. Theory and applications. · Zbl 0890.42004
[6] Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. · Zbl 0971.42023 · doi:10.1090/memo/0697
[7] Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628 – 666. · Zbl 0683.42031 · doi:10.1137/1031129
[8] A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), no. 5, 720 – 731. · Zbl 0486.46027 · doi:10.1063/1.525426
[9] A. J. E. M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res. 43 (1988), no. 1, 23 – 69. · Zbl 0653.94002
[10] A.J.E.M. Janssen, Zak transforms with few zeros and the tie, preprint. · Zbl 1027.42025
[11] John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. · Zbl 0185.11502
[12] A. M. Odlyzko and B. Poonen, Zeros of polynomials with 0,1 coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317 – 348. · Zbl 0814.30006
[13] Amos Ron and Zuowei Shen, Frames and stable bases for shift-invariant subspaces of \?\(_{2}\)(\?^{\?}), Canad. J. Math. 47 (1995), no. 5, 1051 – 1094. · Zbl 0838.42016 · doi:10.4153/CJM-1995-056-1
[14] Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in \?\(_{2}\)(\?^{\?}), Duke Math. J. 89 (1997), no. 2, 237 – 282. · Zbl 0892.42017 · doi:10.1215/S0012-7094-97-08913-4
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