A classification of anisotropic Besov spaces

Cheshmavar, Jahangir; Führ, Hartmut

doi:10.1016/j.acha.2019.04.006

Cited by 16 publications

(63 citation statements)

References 25 publications

(52 reference statements)

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“…The proof of Theorem 5.1 involves writing A in real Jordan form, and relating the action of A on balls to the action of corresponding matrices with positive eigenvalues on the same balls, and then to diagonal matrices. Lemma 5.3 is essentially a reformulation of [4,Lemma 6.7]. We give its proof for the sake of completeness.…”

Section: Corollary 52 Suppose Thatmentioning

confidence: 95%

Simultaneous dilation and translation tilings of $\mathbb R^n$

Bownik,

Speegle

2021

Preprint

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We solve the wavelet set existence problem. That is, we characterize the fullrank lattices Γ ⊂ R n and invertible n × n matrices A for which there exists a measurable set W such that {W + γ : γ ∈ Γ} and {A j (W ) : j ∈ Z} are tilings of R n . The characterization is a non-obvious generalization of the one found by Ionascu and Wang [12], which solved the problem in the case n = 2. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues λ satisfy |λ| ≥ 1. As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is ≥ 1.which forms an orthogonal basis for L 2 (R n ), where B is the transpose of A and Γ * is the dual lattice of Γ. In dimension 2 and higher, it is an open problem to determine for which pairs (B, Γ * ), there exists a (B, Γ * ) orthonormal wavelet, see [3,23,25]. The current paper provides many new examples of pairs for which such wavelets exist. For all cases currently known, when there exists an orthonormal wavelet, there also exists an (A, Γ) wavelet set. The authors conjecture that this is true in general:Conjecture 1.2. For each pair (B, Γ * ) such that there exists a (B, Γ * ) orthonormal wavelet, there exists an (A, Γ) wavelet set, where B is the transpose of A and Γ * is the dual lattice of Γ.

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Section: Corollary 52 Suppose Thatmentioning

confidence: 95%

Simultaneous dilation and translation tilings of $\mathbb R^n$

Bownik,

Speegle

2021

Preprint

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“…The Besov covering B(G) of a stratified Lie group (R n , * G ) fits in a larger class of coverings investigated in [6] known as inhomogeneous covering induced by an expansive matrix. An expansive matrix A is a matrix such that all its eigenvalues have norm strictly greater than one.…”

Section: Remarkmentioning

confidence: 99%

“…Written in this framework, we could use [6,Lemma 6.1 (b)] to derive the first statement in Proposition 5.4.…”

Section: Remarkmentioning

confidence: 99%

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$$\alpha $$-Modulation Spaces for Step Two Stratified Lie Groups

Berge

2022

J Geom Anal

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We define and investigate $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }({\mathbb {R}}^n)$$ M p , q s , α ( R n ) that act as intermediate spaces between the modulation spaces ($$\alpha = 0$$ α = 0 ) in time-frequency analysis and the Besov spaces ($$\alpha = 1$$ α = 1 ) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases $$\alpha = 0,1$$ α = 0 , 1 where the spaces $$M_{p,q}^{s}(G)$$ M p , q s ( G ) and $${\mathcal {B}}_{p,q}^{s}(G)$$ B p , q s ( G ) have non-standard translation and dilation symmetries. Moreover, we show that the spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $${\mathcal {Q}}(G)$$ Q ( G ) underlying the $$\alpha = 0$$ α = 0 case $$M_{p,q}^{s}(G)$$ M p , q s ( G ) allows for the existence of geometric embeddings $$\begin{aligned} F:M_{p,q}^{s}({\mathbb {R}}^k) \longrightarrow {} M_{p,q}^{s}(G), \end{aligned}$$ F : M p , q s ( R k ) ⟶ M p , q s ( G ) , as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.

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“…We aim to derive necessary and sufficient conditions for the embedding statements Ḃα p,r (A) ֒→ W n,q , B α p,r (A) ֒→ W n,q . Our paper rests on a description of anisotropic Besov spaces as decomposition spaces, established in [3], and general embedding theorems for decomposition spaces proved in [10]. We review the pertinent definitions and results in Section 2, and make some observations that will allow to reduce the discussion of general expansive matrices A to certain normal forms.…”

Section: Introductionmentioning

confidence: 99%