SHOP.AGUARDIENTECLOTHING.COM Books > Functional Analysis > Affine density in wavelet analysis by Gitta Kutyniok

Affine density in wavelet analysis by Gitta Kutyniok

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.

Show description

Read or Download Affine density in wavelet analysis PDF

Similar functional analysis books

A panorama of harmonic analysis

Tracing a course from the earliest beginnings of Fourier sequence via to the most recent learn A landscape of Harmonic research discusses Fourier sequence of 1 and a number of other variables, the Fourier remodel, round harmonics, fractional integrals, and singular integrals on Euclidean area. The climax is a attention of rules from the viewpoint of areas of homogeneous kind, which culminates in a dialogue of wavelets.

Real and Functional Analysis

This e-book introduces most crucial points of contemporary research: the idea of degree and integration and the idea of Banach and Hilbert areas. it's designed to function a textual content for first-year graduate scholars who're already conversant in a few research as given in a ebook just like Apostol's Mathematical research.

Lineare Funktionalanalysis: Eine anwendungsorientierte Einführung

Die lineare Funktionalanalysis ist ein Teilgebiet der Mathematik, das Algebra mit Topologie und research verbindet. Das Buch führt in das Fachgebiet ein, dabei bezieht es sich auf Anwendungen in Mathematik und Physik. Neben den vollständigen Beweisen aller mathematischen Sätze enthält der Band zahlreiche Aufgaben, meist mit Lösungen.

Additional resources for Affine density in wavelet analysis

Sample text

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).

Download PDF sample

Rated 4.97 of 5 – based on 35 votes