Roots of Complex Polynomials and Weyl-heisenberg Frame Sets
نویسندگان
چکیده
A Weyl-Heisenberg frame for L2(R) is a frame consisting of modulates Embg(t) = e 2πimbtg(t) and translates Tnag(t) = g(t − na), m,n ∈ Z, of a fixed function g ∈ L2(R), for a, b ∈ R. A fundamental question is to explicitly represent the families (g, a, b) so that (EmbTnag)m,n∈Z is a frame for L2(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 χE for which (EmTnχE)m,n∈Z is a frame for L2(R) is equivalent to classifying the integer sets {n1 < n2 < · · · < nk} so that f(z) = ∑k j=1 z ni does not have any zeroes on the unit circle in the plane.
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