By Gitta Kutyniok

In wavelet research, abnormal wavelet frames have lately come to the leading edge of present examine as a result of questions about the robustness and balance of wavelet algorithms. an important hassle within the examine of those structures is the hugely delicate interaction among geometric homes of a series of time-scale indices and body homes of the linked wavelet systems.

This quantity offers the 1st thorough and complete remedy of abnormal wavelet frames via introducing and making use of a brand new concept of affine density as a powerful device for studying the geometry of sequences of time-scale indices. some of the effects are new and released for the 1st time. issues 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 houses of abnormal wavelet frames.

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**Additional info for Affine density in wavelet analysis**

**Example text**

L ⊆ A be given. The deﬁnition of aﬃne density for multiple sequences can be reduced to a simpler form using the disjoint L union Λ = =1 Λ of the sequences Λ1 , . . , ΛL . Employing this new sequence we obtain D+ ({Λ }L=1 ) = D+ (Λ) and D− ({Λ }L=1 ) = D− (Λ). 1 Deﬁnitions for 23 (b) Let Λ1 , . . , ΛL ⊆ A with associated weight functions w : Λ → R+ = 1, . . , L be given. Then L L − − D (Λ , w ) ≤ D ({(Λ , w )}L=1 ) ≤ D ({(Λ , w + )}L=1 ) ≤ =1 D+ (Λ , w ). =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.

SL ⊆ R+ with associated weight functions w : S → R+ = 1, . . , L be given. Then L L D− (S , w ) ≤ D− ({(S , w )}L=1 ) ≤ D+ ({(S , w )}L=1 ) ≤ =1 D+ (S , w ). =1 The following result shows that this density is robust against perturbations of the sequence of scale indices. 14. Let S ⊆ R+ , w : S → R+ , and ε > 0. For each S˜ = {sδs : ε ε s ∈ S, δs ∈ [e− 2 , e 2 ]} equipped with a weight function v : S˜ → R+ deﬁned by ˜ v) and D+ (S, w) = D+ (S, ˜ v). v(sδs ) = w(s), we have D− (S, w) = D− (S, Proof.

X2 Since Φ is an isomorphism, we obtain lim sup sup #(Λ ∩ Kh (x, y)) h→∞ (x,y)∈A µA (Kh ) #(Φ(Φ−1 (Λ)) ∩ (x, y) Φ(Qh )) h2 (x,y)∈A = lim sup sup h→∞ #(Φ(Φ−1 (Λ) ∩ Qh (x, y))) h2 (x,y)∈A = lim sup sup h→∞ #(Φ−1 (Λ) ∩ Qh (x, y)) h2 (x,y)∈A = lim sup sup h→∞ = 1 . b ln a 32 3 Weighted Aﬃne Density = D+ (Φ−1 (Λ)) 1 . 3. A similar argument proves the remaining claim. , they are equal in a qualitative sense concerning the upper density. 4. 5 is used), we omit them. 12. If Λ ⊆ A and w : Λ → R+ , then the following conditions are equivalent.