The frame set conjecture for B-splines B n , n ≥ 2, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form ab = r, where r is a rational number smaller than one and a and b denote the sampling and modulation rates, respectively, has infinitely many pieces, located around b = 2, 3, . . . , not belonging to the frame set of the nth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines B n , n ≥ 2.2010 Mathematics Subject Classification. Primary 42C15. Secondary: 42A60
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 ∈ 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 ∈ R. The set of all dual windows h ∈ 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 ∈ C n (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 / ∈ C n+1 (R) then, in general, a dual window h in C n+1 (R) does not exist.2010 Mathematics Subject Classification. Primary 42C15. Secondary: 42A60
We study Gabor frames of the form e 2πibm• g(• − ak) m,k∈Z generated by n times differentiable windows g that are non-zero on an open interval of length L > 0 with translation parameter a = L/2 and modulation parameter b ∈ (0, 2/L). We first review recent explicit constructions of all dual windows with sufficiently short support by the authors. We then show that the obtainable smoothness of dual windows depends on the location of singularities of g. Our proof yields an explicit construction procedure of smooth dual windows once the singularities of g avoid the specified locations.
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