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 because of questions in regards to the robustness and balance of wavelet algorithms. a huge hassle within the learn of those structures is the hugely delicate interaction among geometric homes of a chain of time-scale indices and body homes of the linked wavelet systems.

This quantity presents the 1st thorough and accomplished remedy of abnormal wavelet frames by means of introducing and using a brand new thought of affine density as a powerful software 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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**Additional info for Affine density in wavelet analysis**

**Example text**

1. Recall that the boxes Qh were chosen to be Qh = 28 3 Weighted Aﬃne Density [e− 2 , e 2 ) × [− h2 , h2 ). Then, for h > 0 and (x, y) ∈ A, we deﬁne the boxes Qh (x, y) by h h Qh (x, y) = (x, y) Qh = {(xa, y + xb) : a ∈ [e− 2 , e 2 ), b ∈ [− h2 , h2 )} h h We remark that in this situation the sets Qh (x, y) remain rectangles, thus the geometry of the measuring boxes is easier in this sense. A left-invariant Haar measure for the group A is given by µA = dx x2 dy. Hence the volume of a box Qh (x, y) equals µA (Qh (x, y)) = µA (Qh ) = h h 2 e2 h −2 h e− 2 h h dx dy = h(e 2 − e− 2 ).

I) {Qh (ejh , khejh )}j,k∈Z is a disjoint covering of A. (ii) Any set Qrh (x, y) intersects at most jh r(eh( r+5 r+3 ) 2 −e−h( 2 ) ) h e −1 + 2(r + 2) sets of jh the form Qh (e , khe ). (iii) Any set Qrh (x, y) contains at least r(eh( r−5 r−3 ) 2 −e−h( 2 ) ) h e −1 − 2(r + 2) disjoint sets of the form Qh (ejh , khejh ). Proof. (i) Fix any (x, y) ∈ A. 3 The Notion of Aﬃne Density by Sun and Zhou 29 ln x = jh + ln a. Further there exists a unique k ∈ Z and some b ∈ [− h2 , h2 ) such that y = ejh (kh + b).

L ∈ L2 (R), and let Λ1 , . . , ΛL ⊆ A with disL L −1 + ) < ∞. If joint union denoted by Λ = =1 Λ . Suppose that D ( =1 Λ L 2 − =1 W(ψ , Λ ) possesses a lower frame bound for L (R), then D (Λ) > 0. 6, we give a further interpretation of ﬁnite upper aﬃne density in terms of the following deﬁnition. 3. 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.

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