Some counterexamples in the theory of Weyl-Heisenberg frames

Janssen, Ajem Guido

doi:10.1109/18.485730

Cited by 7 publications

(4 citation statements)

References 7 publications

(3 reference statements)

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“…One of the central motives of this article is hinted by the Daubechies conjecture [3, p. 981] which assumes that Gabor system is a frame for all αβ < 1 whenever g is a positive function with positive Fourier transform. This conjecture has been disproved by Janssen [8], yet in all known examples of functions which generates a Gabor system for all αβ < 1 we encounter some kind of positivity.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Gabor frames for rational functions

Belov¹,

Kulikov

Lyubarskii³

2022

Invent. math.

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We study the frame properties of the Gabor systems $$\begin{aligned} {\mathfrak {G}}(g;\alpha ,\beta ):=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in {\mathbb {Z}}}. \end{aligned}$$ G ( g ; α , β ) : = { e 2 π i β m x g ( x - α n ) } m , n ∈ Z . In particular, we prove that for Herglotz windows g such systems always form a frame for $$L^2({\mathbb {R}})$$ L 2 ( R ) if $$\alpha ,\beta >0$$ α , β > 0 , $$\alpha \beta \le 1$$ α β ≤ 1 . For general rational windows $$g\in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) we prove that $${\mathfrak {G}}(g;\alpha ,\beta )$$ G ( g ; α , β ) is a frame for $$L^2({\mathbb {R}})$$ L 2 ( R ) if $$0<\alpha ,\beta $$ 0 < α , β , $$\alpha \beta <1$$ α β < 1 , $$\alpha \beta \not \in {\mathbb {Q}}$$ α β ∉ Q and $${\hat{g}}(\xi )\ne 0$$ g ^ ( ξ ) ≠ 0 , $$\xi >0$$ ξ > 0 , thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $$L^2({\mathbb {R}})$$ L 2 ( R ) .

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Section: Introduction and Main Resultsmentioning

confidence: 93%

Gabor frames for rational functions

Belov¹,

Kulikov

Lyubarskii³

2022

Invent. math.

View full text Add to dashboard Cite

show abstract

“…One of the central motives of the article is hinted by the Daubechies conjecture [3, p. 981] which assumes that Gabor system is a frame for all αβ < 1 whenever g is positive function with positive Fourier transform. This conjecture has been disproved by Janssen [8], yet in all known examples of functions which generates a Gabor system for all αβ < 1 we encounter some kind of positivity.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Gabor frames for rational functions

Belov¹,

Kulikov²,

Lyubarskii³

2021

Preprint

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We study the frame properties of the Gabor systemsIn particular, we prove that for Herglotz windows g such systems always form a frame for L 2 (R) if α, β > 0, αβ ≤ 1. For general rational windows g ∈ L 2 (R) we prove that G(g; α, β) is a frame for L 2 (R) if 0 < α, β, αβ < 1, αβ ∈ Q and ĝ(ξ) = 0, ξ > 0, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L 2 (R).

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“…(i) γ(x) is piecewise continuous, (ii) card (supp γ(x)) ≤ (r + 1)M, where according to (18) supp γ(x) = {j : γ 0,j (x) ≡ 0} ⊆ {j :…”

Section: Gabor Frames With Totally Positive Functionsmentioning

confidence: 99%

“…The example of the Gaussian lead Daubechies to conjecture that F (g) = {(α, β) ∈ R 2 + : αβ < 1} whenever g is a positive function in L 1 with positive Fourier transform in L 1 [9, p. 981]. This conjecture was disproved in [18].…”

Section: Introductionmentioning

confidence: 99%

Gabor frames and totally positive functions

Gröchenig

Stöckler

2013

Duke Math. J.

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Abstract. Let g be a totally positive function of finite type, i.e.,ĝ(ξ) = M ν=1 (1 + 2πiδ ν ξ) −1 for δ ν ∈ R and M ≥ 2. Then the set {e 2πiβlt g(t − αk) : k, l ∈ Z} is a frame for L 2 (R), if and only if αβ < 1. This result is a first positive contribution to a conjecture of I. Daubechies from 1990. So far the complete characterization of lattice parameters α, β that generate a frame has been known for only six window functions g. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.

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Some counterexamples in the theory of Weyl-Heisenberg frames

Cited by 7 publications

References 7 publications

Gabor frames for rational functions

Gabor frames for rational functions

Gabor frames for rational functions

Gabor frames and totally positive functions

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