TY - JOUR
T1 - A density theorem for time-continuous filter banks
AU - Janssen, A.J.E.M.
PY - 1998
Y1 - 1998
N2 - We show in this paper the following result. When a > 0 and gnm = gm(· - na), n, m e Z, is a frame for L2(), where each gm e L2() is localized in the frequency-domain around a point b m e , then ( = 1. Here is the asymptotic lower bound of the average number of bm in a symmetric interval around 0 as the interval length tends to 8. As a particular case it is found that a multi-window Gabor-type system gp(t - na) exp(2pimbt) with n, m e Z, p = 0, …, P - 1, can generate a frame for L2() only if P(ab)-1 = 1. The main result of this paperis based on the Ron-Shen criterion in the frequency-domain for duality of two shift- invariant systems of functions, combined with an amplification of Janssen's elementary proof of the fact that a Weyl-Heisenberg system g(t - na)exp(2pimbt) can generate a framefor L2() only if (ab)-1 = 1.
AB - We show in this paper the following result. When a > 0 and gnm = gm(· - na), n, m e Z, is a frame for L2(), where each gm e L2() is localized in the frequency-domain around a point b m e , then ( = 1. Here is the asymptotic lower bound of the average number of bm in a symmetric interval around 0 as the interval length tends to 8. As a particular case it is found that a multi-window Gabor-type system gp(t - na) exp(2pimbt) with n, m e Z, p = 0, …, P - 1, can generate a frame for L2() only if P(ab)-1 = 1. The main result of this paperis based on the Ron-Shen criterion in the frequency-domain for duality of two shift- invariant systems of functions, combined with an amplification of Janssen's elementary proof of the fact that a Weyl-Heisenberg system g(t - na)exp(2pimbt) can generate a framefor L2() only if (ab)-1 = 1.
U2 - 10.1016/S1874-608X(98)80020-4
DO - 10.1016/S1874-608X(98)80020-4
M3 - Article
SN - 1874-608X
VL - 7
SP - 513
EP - 523
JO - Wavelet Analysis and its Applications
JF - Wavelet Analysis and its Applications
ER -