Download e-book for kindle: Affine density in wavelet analysis by Gitta Kutyniok

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.

Show description

Read Online or Download Affine density in wavelet analysis PDF

Similar functional analysis books

Read e-book online Calculus 3 PDF

This e-book, the 3rd of a three-volume paintings, is the outgrowth of the authors' adventure instructing calculus at Berkeley. it's all for multivariable calculus, and starts off with the mandatory fabric from analytical geometry. It is going directly to disguise partial differention, the gradient and its functions, a number of integration, and the theorems of eco-friendly, Gauss and Stokes.

Download e-book for iPad: Calculus of variations and harmonic maps by Hajime Urakawa

This publication presents a large view of the calculus of diversifications because it performs a vital function in quite a few components of arithmetic and technology. Containing many examples, open difficulties, and routines with entire recommendations, the e-book will be compatible as a textual content for graduate classes in differential geometry, partial differential equations, and variational tools.

Get Taylor Coefficients and Coefficient Multipliers of Hardy and PDF

This booklet presents a scientific evaluate of the idea of Taylor coefficients of capabilities in a few classical areas of analytic features and particularly of the coefficient multipliers among areas of Hardy style. supplying a complete reference consultant to the topic, it's the first of its style during this zone.

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.

Download PDF sample

Affine density in wavelet analysis by Gitta Kutyniok


by Paul
4.3

Rated 4.37 of 5 – based on 21 votes