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

We consider Gabor frames {e^2πibm·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.