Affine density in wavelet analysis by Gitta Kutyniok PDF

By Gitta Kutyniok

ISBN-10: 354072916X

ISBN-13: 9783540729167

ISBN-10: 3540729496

ISBN-13: 9783540729495

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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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.

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Affine density in wavelet analysis by Gitta Kutyniok

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