Affine density in wavelet analysis by Gitta Kutyniok

By Gitta Kutyniok

In wavelet research, abnormal wavelet frames have lately come to the vanguard of present examine as a result of questions about the robustness and balance of wavelet algorithms. a massive hassle within the learn 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 presents the 1st thorough and entire remedy of abnormal wavelet frames via introducing and utilising a brand new suggestion of affine density as a powerful device for studying the geometry of sequences of time-scale indices. a number of the effects are new and released for the 1st time. issues contain: qualitative and quantitative density stipulations for life of abnormal wavelet frames, non-existence of abnormal co-affine frames, the Nyquist phenomenon for wavelet structures, and approximation houses of abnormal wavelet frames.

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1 These inequalities may be strict, for instance, consider Λ1 = {(2j , k)}j≥0,k∈Z and Λ2 = {(2j , k)}j<0,k∈Z and w1 = w2 = 1, where L = 2. The best known class of wavelet systems are the classical affine systems, which are W(ψ, {(aj , bk)}j,k∈Z ) with a > 1, b > 0, and ψ ∈ L2 (R). For Gabor 1 systems G(g, aZ×bZ) with a, b > 0, the Beurling density ab of the lattice aZ× bZ is a ubiquitous constant in a variety of formulas. For example, if G(g, aZ × bZ) is a tight frame for L2 (R) and g 2 = 1, then the frame bounds are 1 .

We will say that a set K ⊆ A is affinely h-separated if (a, b) = (c, d) ∈ K =⇒ Qh (a, b) ∩ Qh (c, d) = ∅. Before proving a characterization of finite upper affine density in terms of this notion, we require the following technical lemma. 4. Let h > 0 be given. If Qh (x, y) ∩ Qh (a, b) = ∅, then (x, y) ∈ Q h2 (a, b). 2he 38 4 Qualitative Density Conditions Proof. Suppose that (c, d) ∈ Qh (x, y) ∩ Qh (a, b). Then we would have (c, d) = (x, y)(t, u) = (a, b)(r, s) for some (t, u), (r, s) ∈ Qh . Therefore, (a, b)−1 · (x, y) = (r, s) · (t, u)−1 = ( rt , st − tu) ∈ Q h 2he 2 so (x, y) ∈ Q h 2he 2 , (a, b).

Recall that in this case we only consider m ∈ Z\{0}. Therefore there exists ε > 0 such that Kh (s) ∩ (Kh (s) − m b ) = ∅ for all s ∈ S with s < ε and m ∈ Z\{0}. This shows that we only need to consider those s ∈ S with s ≥ ε. Then there exists a C < ∞ such that h h 3(2sb(e 2 − e− 2 ) + 1) ≤ Cs(e 2 − e− 2 ) for all s ∈ S, s ≥ ε. h h h h Using these observations, we obtain I2 (h) ≤ 1 b s∈S h h w(s) Cs(e 2 − e− 2 ) s h C h = (e 2 − e− 2 ) b 2 ˆ |ψ(ξ)| dξ Kh (s) 2 ˆ |ψ(ξ)| dξ. 10) 54 4 Qualitative Density Conditions It follows easily from D+ (S, w) < ∞ that there exists an N < ∞ such that #w {s ∈ S : x ∈ Kh (s)} ≤ N for all x ∈ R.

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