By Gitta Kutyniok
In wavelet research, abnormal wavelet frames have lately come to the leading edge of present learn because of questions about the robustness and balance of wavelet algorithms. an enormous hassle within the examine of those structures is the hugely delicate interaction among geometric houses of a chain of time-scale indices and body houses of the linked wavelet systems.
This quantity offers the 1st thorough and complete remedy of abnormal wavelet frames via introducing and applying a brand new idea of affine density as a powerful device for reading the geometry of sequences of time-scale indices. a few of the effects are new and released for the 1st time. subject matters comprise: qualitative and quantitative density stipulations for lifestyles of abnormal wavelet frames, non-existence of abnormal co-affine frames, the Nyquist phenomenon for wavelet platforms, and approximation homes of abnormal wavelet frames.
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Extra info for Affine density in wavelet analysis
We will transfer the deﬁnition of Beurling density for sequences in R2 to sequences in the aﬃne group and extend it in order to allow multiple weighted sequences. Before we can state the deﬁnition of aﬃne density, we ﬁrst require some notation. For h > 0, we let Qh denote a ﬁxed family of increasing, exhaustive neighborhoods of the identity element e = (1, 0) in A. For simplicity, we will take h h Qh = [e− 2 , e 2 ) × [− h2 , h2 ). , 22 3 Weighted Aﬃne Density Qh (x, y) = (x, y) · Qh = (xa, ay + b) : a ∈ [e− 2 , e 2 ), b ∈ [− h2 , h2 ) .
1 1 (i) Suppose that yn → 0 as n → ∞. Then the function ψ ∈ L2 (R) deﬁned by ψˆ = n∈N 1 1 χ −1 n yn [e 2 ,e 2 ) satisﬁes ψˆ ∈ WR∗ (L∞ , L2 ). (ii) Suppose that yn → ∞ as n → ∞. Then the function ψ ∈ L2 (R) deﬁned by 1 1 ψˆ = √ χ −1 n yn yn [e 2 ,e 2 ) n∈N satisﬁes ψˆ ∈ WR∗ (L∞ , L2 ). Proof. (i) Suppose that yn → 0 as n → ∞. It is easy to check that ψˆ ∈ L2 (R), hence ψ ∈ L2 (R). We next observe that for each k ∈ Z and x ∈ R+ , we have 1 1 1 1 ek [e− 2 , e 2 ) ∩ x[e− 2 , e 2 ) = ∅ if and only if ln x − 1 ≤ k ≤ ln x + 1.
The following lemma is similar to a result by Christensen, Deng, and Heil [22, Lem. 3] for Gabor systems, where here we make use of the Bergman transform instead of the Bargmann transform and choose a diﬀerent function η. 8. Let ψ ∈ L2 (R), and deﬁne η ∈ L2A (R) by ηˆ(ξ) = 2ξ e−ξ , 0, ξ ≥ 0, ξ < 0. For each δ > 1, there exists a constant Cδ > 0 such that for every (p, q), (a, b) ∈ A, 2 | σ(p, q)η, σ(a, b)ψ | ≤ Cδ ψ, σ(x, y)η Qδ ((a,b)−1 ·(p,q)) 2 dµA (x, y). 40 4 Qualitative Density Conditions Proof.
Affine density in wavelet analysis by Gitta Kutyniok