By Gitta Kutyniok

In wavelet research, abnormal wavelet frames have lately come to the leading edge of present study as a result of questions about the robustness and balance of wavelet algorithms. an enormous hassle within the learn 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 finished therapy of abnormal wavelet frames via introducing and making use of a brand new suggestion of affine density as a powerful instrument for interpreting the geometry of sequences of time-scale indices. some 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 homes of abnormal wavelet frames.

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L W(ψ , S × {0}). = =1 Consider ﬁrst the case of pure dilations. Note that D− ( =1 S ×{0}) = 0. 1. L Suppose on the other hand that D+ ( =1 S × {0}) < ∞. 4(i), D (S × {0}) < ∞ for = 1, . . , L. Fix ∈ {1, . . , L}. If (c, 0) ∈ (S × h h −1 ·(c, 0) ∈ Qh . Hence e− 2 ≤ xc < e 2 , {0})−1 ∩Qh (x, y), then ( xc , − xy c ) = (x, y) cy xy c c cy h −h h h x 1 1 −1 so − 2 e < − x = − c x x < 2 e . Therefore ( c , − x ) = ( x , y) · ( c , 0) ∈ Qheh , so ( 1c , 0) ∈ Qheh ( x1 , y). Thus L sup #((S × {0})−1 ∩ Qh (x, y)) ≤ sup #((S × {0}) ∩ Qheh (x, y)) < ∞, (x,y)∈A (x,y)∈A so D+ ((S × {0})−1 ) < ∞ for all = 1, .

SL ⊆ R+ be given. ,L is not a frame for L2 (R). ,L is not a frame for L2 (R). Proof. ,L W(ψ , S × {0}). = =1 Consider ﬁrst the case of pure dilations. Note that D− ( =1 S ×{0}) = 0. 1. L Suppose on the other hand that D+ ( =1 S × {0}) < ∞. 4(i), D (S × {0}) < ∞ for = 1, . . , L. Fix ∈ {1, . . , L}. If (c, 0) ∈ (S × h h −1 ·(c, 0) ∈ Qh . Hence e− 2 ≤ xc < e 2 , {0})−1 ∩Qh (x, y), then ( xc , − xy c ) = (x, y) cy xy c c cy h −h h h x 1 1 −1 so − 2 e < − x = − c x x < 2 e . Therefore ( c , − x ) = ( x , y) · ( c , 0) ∈ Qheh , so ( 1c , 0) ∈ Qheh ( x1 , y).

6. 5. If Λ ⊆ A, then the following conditions are equivalent. (i) D+ (Λ) < ∞. (ii) There exists h > 0 such that Λ can be written as a ﬁnite union of sequences Λ1 , . . , ΛN , each of which is aﬃnely h-separated. (iii) For every h > 0, Λ can be written as a ﬁnite union of sequences Λ1 , . . , ΛN , each of which is aﬃnely h-separated. Proof. (i) ⇒ (iii). Assume that D+ (Λ) < ∞, and let h > 0 be given. 6, we have M = sup(x,y)∈A #(Λ ∩ Qh (x, y)) < ∞. Fix any (a, b) ∈ Λ. 4 that (c, d) ∈ Q h2 (a, b).