Affine Density in Wavelet Analysis (Lecture Notes in by Gitta Kutyniok

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By Gitta Kutyniok

This quantity offers an intensive and accomplished therapy of abnormal wavelet frames. It introduces and employs a brand new inspiration of affine density as a powerful device for interpreting the geometry of sequences of time-scale indices. assurance contains non-existence of abnormal co-affine frames, the Nyquist phenomenon for wavelet platforms, and approximation houses of abnormal wavelet frames.

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10), we obtain I2 (h) ≤ h C h (e 2 − e− 2 ) b 2 ˆ |ψ(ξ)| dξ ≤ w(s) Kh (s) s∈S h C h (e 2 − e− 2 ) N ψˆ b 2 2 < ∞. 8). Thus (ii) holds. (ii) ⇒ (i). Suppose that (ii) holds. Towards a contradiction assume that we have D+ ({(S , w )}L=1 ) = ∞. 17, there exists 0 ∈ {1, . . , L} such that D+ (S 0 , w 0 ) = ∞. 23, to obtain a contradiction it suffices to show that there exists ψ ∈ L2 (R) with ψˆ ∈ WR∗ (L∞ , L2 ) such that for some h > 0, I 0 (h) = 1 b0 s∈S w 0 (s) s 0 Kh (s)∩(Kh (s)− bm 0 m∈Z 2 ˆ |ψ(ξ)| dξ = ∞.

By the disjointization principle of Feichtinger and Gr¨ obner [51, Lem. 9], it follows that Λ can be divided into at most M N subsequences Λ1 , . . , ΛM N such that for each fixed i, the sets Qh (a, b) with (a, b) ∈ Λi are disjoint, or in other words, Λi is affinely h-separated. h (ii) ⇒ (i). Assume that Λ = Λ1 ∪· · ·∪ΛN with each Λi affinely h-separated. δ Fix δ so that 1 < 2δe 2 < h, and suppose that two points (a, b) and (c, d) of some Λi were both contained in some Qδ (x, y). 4, we would have (x, y) ∈ Q δ2 (a, b) ⊆ Qh (a, b) and (x, y) ∈ Q δ2 (c, d) ⊆ Qh (c, d).

1 This proves (ii). 1, the notion of affine density can also be defined by employing a group isomorphic to A. Let A = R+ × R denote this group, which is endowed with multiplication given by (a, b) (x, y) = (ax, b + ay). To distinguish this group multiplication from the one associated with the group A, we use the operator sign here. The identity element of A is e = (1, 0), and the inverse of an element (a, b) ∈ A is given by (a, b)−1 = ( a1 , − ab ). To construct a wavelet system adapted to this group, let σ ˜ be the unitary representation of A on L2 (R) defined by (˜ σ (a, b)ψ)(x) = √1 ψ( x−b ).

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