On the Weyl-Heisenberg frames generated by simple functions
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AbstractLet phi(x) = Sigma(infinity)(n=0) c(n)chi(E)(x - n) with {c(n)}(n=0)(infinity) is an element of l(1), and let (phi,a, 1), 0 < a <= 1 be a Weyl-Heisenberg system {e(2 pi imx) phi(x - na): m, n is an element of Z}. We show that if E = [0, 1] (and some modulo extension of E), then (phi, a, 1) is a frame for each 0 < a <= 1 (for certain a, respectively) if and only if the analytic function H(z) = Sigma(infinity)(n=0) c(n)z(n) has no zero on the unit circle {z: vertical bar z vertical bar = 1}. These results extend the case of Casazza and Kalton (2002) [6] that phi(x) = Sigma(k)(i=1) chi([0, 1])(x - n(i)) and a = 1, which brought together the frame theory and the function theory on the closed unit disk. Our techniques of proofs are based on the Zak transform and the distribution of fractional parts of {na}(n is an element of Z). (c) 2011 Elsevier Inc. All rights reserved.
All Author(s) ListHe XG, Lau KS
Journal nameJournal of Functional Analysis
Volume Number261
Issue Number4
Pages1010 - 1027
LanguagesEnglish-United Kingdom
KeywordsAnalytic function; Fractional part; Frame; Modulation; Translation; Zak transform; Zero
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-25-01 at 00:12