Validity of WH-frame bound conditions depends on Lattice parameters

H.G. Feichtinger, A.J.E.M. Janssen

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In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational a>0 we can construct a smooth g¿ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m¿ Zof time-frequency translates of ghas a finite frame upper bound, while for any ß>0 and any rational c> 0 the collection (gnca, mß)n, m¿ Zhas no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g¿ L2(R) which is sufficiently well behaved, there exist ac>0, bc>0 such that (gn a, m b)n, m¿ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g¿ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3- kwith l, k¿ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)- 1with m¿ N and all b, 0 < b <1.
Original languageEnglish
Pages (from-to)104-112
Number of pages9
JournalApplied and Computational Harmonic Analysis
Issue number1
Publication statusPublished - 2000


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