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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An uneven norm is a good convinced sublinear practical p on a true vector house X. The topology generated by means of the uneven norm p is translation invariant in order that the addition is continuing, however the asymmetry of the norm means that the multiplication by way of scalars is continuing in basic terms whilst constrained to non-negative entries within the first argument. dissertation survey Nonlinear Functional Analysis: A First Course

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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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