Gabor windows supported on [ − 1, 1] and dual windows with small support

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

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“…In short, our work can be seen as a a continuation of [7,8], but with the objective of [16] to construct smooth dual pairs of Gabor windows.…”

Section: Results In the Literaturementioning

confidence: 99%

“…In Section 3.4 we discuss optimal smoothness of dual windows and show that the results in Section 3.3, in general, are optimal. In Section 3.5 we minimize the support length of the dual window in the sense of [8] while preserving the optimal smoothness. However, we first introduce some notation used below.…”

Section: Properties Of the Dual Windowsmentioning

confidence: 99%

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

Lemvig

Nielsen

2020

Journal of Approximation Theory

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“…The proof of the necessity of the conditions in Theorem 2.1 is similar to the proof in [2], so we skip this part. On the other hand, it requires much more work to prove that {E mb T na g} m,n∈Z is a frame if the conditions (i)-(iv) are satisfied.…”

Section: General Resultsmentioning

confidence: 99%

“…The proof is quite complicated and is split into several lemmas and intermediate steps. The idea of the proof was gained through the work on the special case with translation parameter a = 1 (see the paper [2]), as well as the observation that the duality condition (3.3) forces a certain behavior of the window g around points x 0 + a for which g(x 0 ) = 0. As further help to understand the idea behind the proof we prove the steps directly in a concrete case, see Example 2.2.…”

Section: Introductionmentioning

confidence: 99%

On Gabor frames generated by sign-changing windows and B-splines

Christensen

Kim

2015

Applied and Computational Harmonic Analysis

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For a class of compactly supported windows we characterize the frame property for a Gabor system {E mb T na g} m,n∈Z , for translation parameters a belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of "curves." For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a, b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gröchenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain "hyperbolic strip."

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“…The literature on Gabor frames has expanded considerably in the last years. In [1] Christensen finds the conditions on the parameters a and b such that (g, a, b) is a frame with a dual generated by a finite linear combination of the translates of g. More recently, in [2], the same author shows that a frame (g, 1, b) always has a compactly supported dual window if g is bounded, supported in [−1, 1] and b ∈]1/2, 1[. Laugesen [9] introduces a method for constructing dual Gabor window functions that are polynomial splines.…”

Section: Introductionmentioning

confidence: 98%

On a necessary condition for B-spline Gabor frames

Prete

2010

Ricerche mat.

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In a previous note K. Grochenig et al. prove that if g is a continuous function with compact support such that the translates of g form a partition of unity, then g cannot generate a Gabor frame for integer values of the frequency shift parameter b greater than 1. We give a simpler proof of this result which applies also to windows g which are neither continuous nor with compact support. Our proof is based on a necessary condition for Gabor frames due to C. E. Heil and D. F. Walnut

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Gabor windows supported on [ − 1, 1] and dual windows with small support

Cited by 18 publications

References 10 publications

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

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

On Gabor frames generated by sign-changing windows and B-splines

On a necessary condition for B-spline Gabor frames

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