Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Czaja, Wojciech; Kutyniok, Gitta; Speegle, Darrin

doi:10.1007/s00041-008-9026-0

Cited by 20 publications

(11 citation statements)

References 22 publications

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“…One result on Beurling dimension that we will use in the main section of this paper is the following [8,9].…”

Section: Beurling Dimensionmentioning

confidence: 98%

Uniform partitions of frames of exponentials into Riesz sequences

Speegle

2008

Journal of Mathematical Analysis and Applications

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The Feichtinger Conjecture, if true, would have as a corollary that for each set $E\subset \T$ and $\Lambda \subset \Z$, there is a partition $\Lambda_1,...,\Lambda_N$ of $\Z$ such that for each $1\le i \le N$, $\{\exp(2\pi i x\lambda): \lambda \in \Lambda_i\}$ is a Riesz sequence. In this paper, sufficient conditions on sets $E\subset \T$ and $\Lambda\subset \R$ are given so that $\{\exp(2\pi i x\lambda) 1_E: \lambda \in \Lambda\}$ can be uniformly partitioned into Riesz sequences

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“…One result on Beurling dimension that we will use in the main section of this paper is the following [8,9].…”

Section: Beurling Dimensionmentioning

confidence: 98%

Uniform partitions of frames of exponentials into Riesz sequences

Speegle

2008

Journal of Mathematical Analysis and Applications

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“…3.3. Given a set Λ ⊂ R d , one defines its upper Beurling dimension to be the infimum of the numbers α for which there exists a constant C such that (3.3) holds (see [CKS08]). Lemma 3.1 thus shows that if E(Λ) is a Bessel system for a measure µ satisfying the condition (3.1) for some α, then the upper Beurling dimension of Λ cannot exceed α.…”

Section: Recall That a System Of Vectors {Fmentioning

confidence: 99%

Fourier frames for singular measures and pure type phenomena

Lev

2018

Proc. Amer. Math. Soc.

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Let µ be a positive measure on R d . It is known that if the space L 2 (µ) has a frame of exponentials then the measure µ must be of "pure type": it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that µ cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each "pure" component of the measure separately may admit such a frame.

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“…If D + (Λ) = D − (Λ), then we say that Λ has uniform Beurling density and we denote this density by D(Λ) (see [4]). Gabardo [8] established a connection between certain convolution inequalities for positive Borel measures in R n and the corresponding notions of Beurling density associated with such measures.…”

Section: Preliminariesmentioning

confidence: 99%

Measure of Self-Affine Sets and Associated Densities

Gabardo

2014

Constr Approx

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Let B be an n × n real expanding matrix and D be a finite subset of R n with 0 ∈ D. The self-affine set K = K(B, D) is the unique compact set satisfying the setvalued equation BK = d∈D (K + d). In the case where card(D) = |det B|, we relate the Lebesgue measure of K(B, D) to the upper Beurling density of the associated measure µ = lim s→∞ ℓ0,...,ℓs−1∈D δ ℓ0+Bℓ1+···+B s−1 ℓs−1 . If, on the other hand, card(D) < |det B| and B is a similarity matrix, we relate the Hausdorff measure H s (K), where s is the similarity dimension of K, to a corresponding notion of upper density for the measure µ.

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Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Cited by 20 publications

References 22 publications

Uniform partitions of frames of exponentials into Riesz sequences

Uniform partitions of frames of exponentials into Riesz sequences

Fourier frames for singular measures and pure type phenomena

Measure of Self-Affine Sets and Associated Densities

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