By Gitta Kutyniok

In wavelet research, abnormal wavelet frames have lately come to the vanguard of present study as a result of questions about the robustness and balance of wavelet algorithms. an immense trouble within the research of those structures is the hugely delicate interaction among geometric houses of a chain of time-scale indices and body homes of the linked wavelet systems.

This quantity offers the 1st thorough and accomplished therapy of abnormal wavelet frames via introducing and making use of a brand new thought of affine density as a powerful software for interpreting 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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6. If Λ ⊆ A and w : Λ → R+ , then the following conditions are equivalent. (i) D+ (Λ, w) < ∞. (ii) There exists h > 0 such that sup(x,y)∈A #w (Λ ∩ Qh (x, y)) < ∞. (iii) For every h > 0 we have sup(x,y)∈A #w (Λ ∩ Qh (x, y)) < ∞. Proof. (i) ⇒ (ii) and (iii) ⇒ (ii) are trivial. (ii) ⇒ (i), (iii). Suppose there exists h > 0 such that M = sup #w (Λ ∩ Qh (x, y)) < ∞. (x,y)∈A For 1 < t < h, we have Qt (x, y) ⊆ Qh (x, y), so sup(x,y)∈A #w (Λ∩Qt (x, y)) < ∞. On the other hand, if t ≥ h then we have t = rh with r ≥ 1.

L be given. Then the upper weighted aﬃne density of {(Λ , w )}L=1 is deﬁned by L =1 D+ ({(Λ , w )}L=1 ) = lim sup sup h→∞ (x,y)∈A #w (Λ ∩ Qh (x, y)) , h2 and the lower weighted aﬃne density of {(Λ , w )}L=1 is D− ({(Λ , w )}L=1 ) = lim inf inf L =1 h→∞ (x,y)∈A #w (Λ ∩ Qh (x, y)) . h2 If D− ({(Λ , w )}L=1 ) = D+ ({(Λ , w )}L=1 ), then we say that {(Λ , w )}L=1 has uniform weighted aﬃne density and denote this density by D({(Λ , w )}L=1 ). If w1 = . . = wL = 1, we write D− ({Λ }L=1 ) and D+ ({Λ }L=1 ).

This requires 2 ln x − h 2 ln x + h ≤j< 2 ln a 2 ln a 2y − xh 2y + xh ≤k< . 2aj b 2aj b and Since terms ±1 are not signiﬁcant in the limit (cf. 3), it suﬃces to observe that xh #(Λ ∩ Qh (x, y)) ≈ b 2 ln x+h 2 ln a −1 2 ln x−h 2 ln a j= 1 . aj By choosing x appropriately, this implies that D− (Λ) = lim inf inf #(Λ ∩ Qh (x, y)) h(e 2 − e− 2 ) h h→∞ (x,y)∈A h(e 2 − e− 2 ) h = lim inf h→∞ h h b(a − 1)h(e 2 − e− 2 ) h h = 1 b(a − 1) and D+ (Λ) = lim sup sup h→∞ (x,y)∈A #(Λ ∩ Qh (x, y)) h(e 2 − e− 2 ) h h(ae 2 − e− 2 ) h = lim sup h→∞ h h h 2 b(a − 1)h(e − h e− 2 ) = a , b(a − 1) which ﬁnishes the proof.