Gabor windows supported on [-1,1] and construction of compactly supported dual windows with optimal frequency localization
Abstract
We consider Gabor frames \e2π i bm · g(·-ak)\m,k ∈ Z with translation parameter a=L/2, modulation parameter b ∈ (0,2/L) and a window function g ∈ Cn(R) supported on [x0,x0+L] and non-zero on (x0,x0+L) for L>0 and x0∈ R. The set of all dual windows h ∈ L2(R) with sufficiently small support is parametrized by 1-periodic measurable functions z. Each dual window h is given explicitly in terms of the function z in such a way that desirable properties (e.g., symmetry, boundedness and smoothness) of h are directly linked to z. We derive easily verifiable conditions on the function z that guarantee, in fact, characterize, compactly supported dual windows h with the same smoothness, i.e., h ∈ Cn(R). The construction of dual windows is valid for all values of the smoothness index n ∈ Z 0 \∞\ and for all values of the modulation parameter b<2/L; since a=L/2, this allows for arbitrarily small redundancy (ab)-1>1. We show that the smoothness of h is optimal, i.e., if g Cn+1(R) then, in general, a dual window h in Cn+1(R) does not exist.