By Gitta Kutyniok

ISBN-10: 354072916X

ISBN-13: 9783540729167

ISBN-10: 3540729496

ISBN-13: 9783540729495

In wavelet research, abnormal wavelet frames have lately come to the leading edge of present study as a result of questions in regards to the robustness and balance of wavelet algorithms. a massive hassle within the examine of those platforms 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 accomplished remedy of abnormal wavelet frames by way of introducing and utilizing 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. themes 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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**Extra resources for Affine density in wavelet analysis**

**Example text**

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, .

17 implies that D+ (S , w ) < ∞ for all = 1, . . , L. For arbitrary functions ψ1 , . . , ψL ∈ L2 (R) with ψˆ1 , . . , ψˆL ∈ WR∗ (L∞ , L2 ), L we have to show that =1 W(ψ , S × b Z, (w , 1)) satisﬁes the LIC. 5 Aﬃne Density and the Local Integrability Condition 53 for all = 1, . . , L and h > 0. For this, ﬁx h > 0, ψ ∈ L2 (R) with ψˆ ∈ WR∗ (L∞ , L2 ), and consider some ∈ {1, . . , L}. For the sake of brevity, we set I(h) = I (h), S = S , w = w , and b = b . We decompose I(h) by I(h) = I1 (h) + I2 (h), where I1 (h) = and I2 (h) = 1 b s∈S 1 b w(s) s s∈S w(s) s 2 ˆ |ψ(ξ)| dξ Kh (s) Kh (s)∩(Kh (s)− m b ) m∈Z\{0} 2 ˆ |ψ(ξ)| dξ.

20. 8. Let φ : R → R be a continuous function with 0 ≤ φ(x) ≤ 1 for all k x ∈ R satisfying that supp(φ) ⊆ K2 , φ|K1 ≡ 1, and k∈Z φ(e ·) ≡ 1 k (this can always be achieved by normalization). Then {φ(e ·)}k∈Z forms a BUPU, since {ek }k∈Z is K2 -dense and relatively separated. Moreover, the 1 k 2 2 is equivalent to norm deﬁned by f = k∈Z ess supx∈R |f (x)φ(e x)| 2 2 2 · WR∗ (L∞ ,L2 ) , since · WR∗ (L∞ ,L2 ) ≤ · ≤ 3 · WR∗ (L∞ ,L2 ) due to the fact that K2 (ek ) ⊆ K1 (ek−1 ) ∪ K1 (ek ) ∪ K1 (ek+1 ) for all k ∈ Z.

### Affine density in wavelet analysis by Gitta Kutyniok

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