Gabor windows supported on [-1,1] and construction of compactly supported dual windows with optimal smoothness

We consider Gabor frames {e(2 pi ibm center dot) g(center dot - ak)}m,k is an element of Z with translation parameter a = L/2, modulation parameter b is an element of (0, 2/L) and a window function g is an element of C-n(R) supported on [x(0), x(0) + L] and non-zero on (x(0), x(0)+L) for L > 0 and x(0) is an element of R. The set of all dual windows h is an element of L-2(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 is an element of C-n(R). The construction of dual windows is valid for all values of the smoothness index n is an element of Z(>= 0) boolean OR {infinity} 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 is not an element of Cn+1(R) then, in general, a dual window h in Cn+1(R) does not exist. (C) 2019 Elsevier Inc, All rights reserved.