ar X iv : 0 90 3 . 49 89 v 4 [ m at h . FA ] 1 9 M ay 2 00 9 GABOR FIELDS AND WAVELET SETS FOR THE HEISENBERG GROUP
نویسنده
چکیده
We study singly-generated wavelet systems on R that are naturally associated with rank-one wavelet systems on the Heisenberg group N . We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N , we give an explicit construction for Parseval frame wavelets that are associated with I . We say that g ∈ L(I×R) is Gabor field over I if, for a.e. λ ∈ I , |λ|g(λ, ·) is the Gabor generator of a Parseval frame for L(R), and that I is a Heisenberg wavelet set if every Gabor field over I is a Parseval frame (mother-)wavelet for L(R). We then show that I is a Heisenberg wavelet set if and only if I is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.
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