Szemerédi regularity lemma: Difference between revisions

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A '''multiresolution analysis (MRA)''' or '''multiscale approximation (MSA)''' is the design method of most of the practically relevant [[discrete wavelet transform]]s (DWT) and the justification for the [[algorithm]] of the [[fast wavelet transform]] (FWT). It was introduced in this context in 1988/89 by [[Stephane Mallat]] and [[Yves Meyer]] and has predecessors in the [[microlocal analysis]] in the theory of [[differential equation]]s (the ''[[ironing method]]'') and the [[pyramid (image processing)|pyramid method]]s of [[image processing]] as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley.
== overwhelmed win over the soul of the house ==


== Definition ==
Kind of near-blind reverence, like pious believers like.<br><br>when cumulative prestige to some level, but also in the hearts quietly sublimation, and no doubt, these alliance members fear for Xiao Yan, also in this [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] sublimation, the more [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ電波ソーラー腕時計レディース] fanatical, in this case, 'medicine' old, who are somewhat surprised, but it did not stop, people always need to have a faith, because this belief, they will have a high degree of cohesion, and Tianfu alliance among the ability to be this belief, apparently with only Xiao Yan<br><br>league carnival, which lasted two weeks later, just gradually subsided, but although unusual member of the excitement [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 電波時計 腕時計] a little crazy, but league executives, but it is still maintained a [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ] calm, overwhelmed [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] win over the soul of the house, for the prestigious Union does has immeasurable benefits, but since the top of this living place, it will inevitably be many slings and arrows, the past trend of strong soul of the house, in
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A ''multiresolution analysis'' of the [[Lp space|space]] <math>L^2(\mathbb{R})</math> consists of a [[sequence]] of nested [[linear subspace|subspaces]]
== 'Come on ==


::<math>\{0\}\dots\subset V_0\subset V_1\subset\dots\subset V_n\subset V_{n+1}\subset\dots\subset L^2(\R)</math>
Tsing Yi with the girls in the juvenile behind.<br><br>'Come on, Ling old, say good-bye.'<br><br>Xiao Yan raised his head, looked before the mountain, he could feel the breath of a large number of forceful, where fast cohesion, and, in the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] other two directions Mountains, also came bursts of spatial fluctuations, it [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] seems Yan Lei II strong family is dispatched<br><br>voice down, Xiao Yan is no longer hesitated, holding the hand Kaoru children, stature flash, is swept [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計] into the sky.<br><br>'Xiao Yan Master, I wish triumphant [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] return!'<br><br>looked back duo Xiao Yan Ling shadow is smile, muttered to himself.<br><br>when Xiao Yan horizon when the two appear in the first mountain, where the air is already shadows scattered, many of the ancient tribe of the strong launch into the sky, forming a neat formation in the sky, the vast surging between breath, like a remote one, that like power and influence, even into the territory of the soul of Emperor [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock] Xiao Yan, are
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that satisfies certain self-similarity relations in time/space and scale/frequency, as well as [[completeness]] and regularity relations.
== and under this acid itch ==


* ''Self-similarity'' in ''time'' demands that each subspace ''V<sub>k</sub>'' is invariant under shifts by [[integer]] [[multiple (mathematics)|multiple]]s of ''2<sup>-k</sup>''. That is, for each <math>f\in V_k,\; m\in\mathbb Z</math> there is a <math>g\in V_k</math> with <math>\forall x\in\mathbb R:\;f(x)=g(x+m2^{-k})</math>.
Gu fine blood into the body, Xiao Yan is covered with the skin immediately harnessed into a bright Jinmang, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html casio 時計] the vast energy in the body as [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン] the storm swept in general open, had exhausted the meridians, almost in an instant, that is once again being gurgle vindictive, quickly filled.<br><br>Moreover, in this energy crisis sweeping the room, Xiao Yan also feel the body's bones, muscles and even the meridians, etc., are among faint heard a strange acid itch, and under this acid itch, as if bones, muscles, etc., are slowly becoming more and more tough and [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 時計 カシオ] strong<br><br>'This is the essence and blood cologne energy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計] inherent in it, really amazing.'<br><br>feel the change in the body, Xiao Yan could not help but come to mind is some amazing promise of 'color', this ability, almost all can, and some higher-order Dan 'medicine' compared to this really worthy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] cologne Void Top-like existence of Warcraft community, just fine blood force, is that it has
* ''Self-similarity'' in ''scale'' demands that all subspaces <math>V_k\subset V_l,\; k<l,</math> are time-scaled versions of each other, with [[Scaling_(geometry)|scaling]] respectively [[Dilation (metric space)|dilation]] factor ''2<sup>l-k</sup>''. I.e., for each <math>f\in V_k</math> there is a <math>g\in V_l</math> with <math>\forall x\in\mathbb R:\;g(x)=f(2^{l-k}x)</math>. If f has limited [[support (mathematics)|support]], then as the support of g gets smaller, the resolution of the ''l''-th subspace is higher than the resolution of the ''k''-th subspace.
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* ''Regularity'' demands that the model subspace ''V<sub>0</sub>'' be generated as the [[linear hull]] ([[algebraic closure|algebraically]] or even [[topologically closed]]) of the integer shifts of one or a finite number of generating functions <math>\phi</math> or <math>\phi_1,\dots,\phi_r</math>. Those integer shifts should at least form a frame for the subspace <math>V_0\subset L^2(\R)</math>, which imposes certain conditions on the decay at infinity. The generating functions are also known as '''[[Wavelet#Scaling_function|scaling functions]]''' or '''[[father wavelets]]'''. In most cases one demands of those functions to be [[piecewise continuous]] with [[compact support]].
== 'But you want me to pan the door completely closed hand ==


* ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be [[dense set|dense]] in <math>L^2(\mathbb{R})</math>, and that they are not too redundant, i.e., their intersection should only contain the zero element.
Means, still useless. '<br><br>'is not it?' Han busy mouth slightly, her eyes are as cold as ice.<br><br>'But you want me to pan the door completely closed hand, non-stick Dan' medicine 'sell this one, is also OK! long as you promise me a proposal.' Xiao Yan suddenly took on the face a little [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio 腕時計 データバンク] sneer, looking across [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 バンド] the Korea idle.<br><br>four hundred and ninetieth chapters conflict (under)<br><br>four hundred and ninetieth chapters conflict<br><br>'proposal? What proposal?'<br><br>hear the words of Xiao Yan, Han busy startled for a while, it seems that it is unexpected, after a long while, just some awareness of road.<br><br>'Since everyone was mixing with' medicine 'division, if genuine Ming dry like [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計] ordinary people, then, is somewhat inconsistent status so [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html casio 腕時計 デジタル] long idle if Korea have the ability, we large available refining' medicine 'approach a division showdown , [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] if I lose, the future nonstick pan door slightest Dan '
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== Important conclusions ==
== we are here.' ==
In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to [[Ingrid Daubechies]].


Assuming the scaling function has compact support, then <math>V_0\subset V_1</math> implies that there is a finite sequence of coefficients <math>a_k=2 \langle\phi(x),\phi(2x-k)\rangle</math> for <math>|k|\leq N</math>, and  <math>a_k=0</math> for <math>|k|>N</math>, such that
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:<math>\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k).</math>
<ul>
 
 
Defining another function, known as '''mother wavelet''' or just '''the wavelet'''
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:<math>\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k).</math>
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One can show that the space <math>W_0\subset V_1</math>, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to <math>V_0</math> inside <math>V_1</math>.{{cn|date=April 2013}} Or put differently, <math>V_1</math> is the [[orthogonal direct sum|orthogonal sum]] (denoted by <math>\oplus</math>) of <math>W_0</math> and <math>V_0</math>. By self-similarity, there are scaled versions <math>W_k</math> of <math>W_0</math> and by completeness one has{{cn|date=April 2013}}
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:<math>L^2(\mathbb R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k,</math>
</ul>
 
thus the set
 
:<math>\{\psi_{k,n}(x)=\sqrt2^k\psi(2^kx-n):\;k,n\in\Z\}</math>
 
is a countable complete [[orthonormal wavelet]] basis in <math>L^2(\R)</math>.
 
==See also==
* [[Multiscale modeling]]
* [[Scale space]]
* [[Wavelet]]
 
{{inline|date=April 2013}}
 
==References==
* {{cite book|first=Charles K.|last=Chui|title=An Introduction to Wavelets|year=1992|publisher=Academic Press|location=San Diego|isbn=0-585-47090-1}}
 
* {{cite book|author1-link=Ali Akansu|first1=A.N.|last1=Akansu|first2=R.A.|last2=Haddad|title=Multiresolution signal decomposition: transforms, subbands, and wavelets|publisher=Academic Press|year=1992|isbn=978-0-12-047141-6}}
 
* {{cite book|author1-link=C. Sidney Burrus|first1=C.S.|last1=Burrus|first2=R.A.|last2=Gopinath|first3=H.|last3=Guo|title=Introduction to Wavelets and Wavelet Transforms: A Primer|publisher=Prentice-Hall|year=1997|isbn=0-13-489600-9}}
 
* {{cite book|first=S.G.|last=Mallat|url=http://www.cmap.polytechnique.fr/~mallat/book.html|title=A Wavelet Tour of Signal Processing|publisher=Academic Press|year=1999|isbn=0-12-466606-X}}
 
== External links ==
*[http://www.docstoc.com/docs/160022503/A-Concise-Introduction-to-Wavelets A Concise Introduction to Wavelets (with  coverage of multiresolution analysis)] by René Puchinger.
 
[[Category:Wavelets]]
[[Category:Time–frequency analysis]]

Latest revision as of 03:15, 1 October 2014

overwhelmed win over the soul of the house

Kind of near-blind reverence, like pious believers like.

when cumulative prestige to some level, but also in the hearts quietly sublimation, and no doubt, these alliance members fear for Xiao Yan, also in this カシオ 腕時計 スタンダード sublimation, the more カシオ電波ソーラー腕時計レディース fanatical, in this case, 'medicine' old, who are somewhat surprised, but it did not stop, people always need to have a faith, because this belief, they will have a high degree of cohesion, and Tianfu alliance among the ability to be this belief, apparently with only Xiao Yan

league carnival, which lasted two weeks later, just gradually subsided, but although unusual member of the excitement カシオ 電波時計 腕時計 a little crazy, but league executives, but it is still maintained a casio 腕時計 メンズ calm, overwhelmed カシオ ソーラー 腕時計 win over the soul of the house, for the prestigious Union does has immeasurable benefits, but since the top of this living place, it will inevitably be many slings and arrows, the past trend of strong soul of the house, in 相关的主题文章：

'Come on

Tsing Yi with the girls in the juvenile behind.

'Come on, Ling old, say good-bye.'

Xiao Yan raised his head, looked before the mountain, he could feel the breath of a large number of forceful, where fast cohesion, and, in the カシオ腕時計 g-shock other two directions Mountains, also came bursts of spatial fluctuations, it カシオ ソーラー 腕時計 seems Yan Lei II strong family is dispatched

voice down, Xiao Yan is no longer hesitated, holding the hand Kaoru children, stature flash, is swept カシオ スタンダード 腕時計 into the sky.

'Xiao Yan Master, I wish triumphant 腕時計 メンズ casio return!'

looked back duo Xiao Yan Ling shadow is smile, muttered to himself.

when Xiao Yan horizon when the two appear in the first mountain, where the air is already shadows scattered, many of the ancient tribe of the strong launch into the sky, forming a neat formation in the sky, the vast surging between breath, like a remote one, that like power and influence, even into the territory of the soul of Emperor casio 腕時計 g-shock Xiao Yan, are 相关的主题文章：

and under this acid itch

Gu fine blood into the body, Xiao Yan is covered with the skin immediately harnessed into a bright Jinmang, casio 時計 the vast energy in the body as カシオ 腕時計 チタン the storm swept in general open, had exhausted the meridians, almost in an instant, that is once again being gurgle vindictive, quickly filled.

Moreover, in this energy crisis sweeping the room, Xiao Yan also feel the body's bones, muscles and even the meridians, etc., are among faint heard a strange acid itch, and under this acid itch, as if bones, muscles, etc., are slowly becoming more and more tough and 時計 カシオ strong

'This is the essence and blood cologne energy カシオ gショック 腕時計 inherent in it, really amazing.'

feel the change in the body, Xiao Yan could not help but come to mind is some amazing promise of 'color', this ability, almost all can, and some higher-order Dan 'medicine' compared to this really worthy カシオ腕時計 g-shock cologne Void Top-like existence of Warcraft community, just fine blood force, is that it has 相关的主题文章：

'But you want me to pan the door completely closed hand

Means, still useless. '

'is not it?' Han busy mouth slightly, her eyes are as cold as ice.

'But you want me to pan the door completely closed hand, non-stick Dan' medicine 'sell this one, is also OK! long as you promise me a proposal.' Xiao Yan suddenly took on the face a little casio 腕時計 データバンク sneer, looking across カシオ 腕時計 バンド the Korea idle.

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'Do not look, we are here.'

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