# 0.6 Regularity, moments, and wavelet system design  (Page 9/13)

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An illustration of the effects of these approximations on a signal is shown in [link] where a signal with a very smooth component (a sinusoid) and a discontinuous component (a square wave) is expanded in awavelet series using samples as the high resolution scaling function coefficients. Notice the effects of projecting onto lower and lowerresolution scales.

If we consider a wavelet system where the same number of scaling function and wavelet moments are set zero and this number is as large as possible,then the following is true [link] , [link] :

Theorem 27 If $m\left(\ell \right)={m}_{1}\left(\ell \right)=0$ for $\ell =1,2,\cdots ,L$ and ${m}_{1}\left(0\right)=0$ , then the ${L}^{2}$ error is

${ϵ}_{3}=\parallel f\left(t\right)-{S}^{j}\left\{f\left(t\right)\right\}{\parallel }_{2}\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}{C}_{3}\phantom{\rule{0.166667em}{0ex}}{2}^{-j\left(L+1\right)},$

where ${C}_{3}$ is a constant independent of $j$ and $L$ , but dependent on $f\left(t\right)$ and the wavelet system.

Here we see that for this wavelet system called a Coifman wavelet system, that using samples as the inner product expansion coefficients is anexcellent approximation. This justifies that using samples of a signal as input to a filter bank gives a proper wavelet analysis. This approximationis also illustrated in [link] and in [link] .

From the previous approximation theorems, we see that a combination of zero wavelet and zero scaling function moments used with samples of thesignal may give superior results to wavelets with only zero wavelet moments. Not only does forcing zero scaling function moments give abetter approximation of the expansion coefficients by samples, it often causesthe scaling function to be more symmetric. Indeed, that characteristic may be more important than the sample approximation in certainapplications.

Daubechies considered the design of these wavelets which were suggested by Coifman [link] , [link] , [link] . Gopinath [link] , [link] and Wells [link] , [link] show how zero scaling function moments give a better approximation of high-resolution scaling coefficients by samples. Tianand Wells [link] , [link] have also designed biorthogonal systems with mixed zero moments with very interesting properties.

The Coifman wavelet system (Daubechies named the basis functions “coiflets") is an orthonormal multiresolution wavelet system with

$\int {t}^{k}\phi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=m\left(k\right)=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,2,\cdots ,L-1$
$\int {t}^{k}\psi \left(t\right)\phantom{\rule{0.166667em}{0ex}}dt={m}_{1}\left(k\right)=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,2,\cdots ,L-1.$

This definition imposes the requirement that there be at least $L-1$ zero scaling function moments and at least $L-1$ wavelet moments in addition to the one zero moment of ${m}_{1}\left(0\right)$ required by orthogonality. This system is said to be of order or degree $L$ and sometime has the additional requirement that the length of the scaling function filter $h\left(n\right)$ , which is denoted $N$ , is minimum [link] , [link] . In the design of these coiflets, one obtains more total zero moments than $N/2-1$ . This was first noted by Beylkin, et al [link] . The length-4 wavelet systemhas only one degree of freedom, so it cannot have both a scaling function moment and wavelet moment of zero (see [link] ). Tian [link] , [link] has derived formulas for four length-6 coiflets. These are:

$h=\left[\frac{-3+\sqrt{7}}{16\sqrt{2}},,,\frac{1-\sqrt{7}}{16\sqrt{2}},,,\frac{7-\sqrt{7}}{8\sqrt{2}},,,\frac{7+\sqrt{7}}{8\sqrt{2}},,,\frac{5+\sqrt{7}}{16\sqrt{2}},,,\frac{1-\sqrt{7}}{16\sqrt{2}}\right],$

or

$h=\left[\frac{-3-\sqrt{7}}{16\sqrt{2}},,,\frac{1+\sqrt{7}}{16\sqrt{2}},,,\frac{7+\sqrt{7}}{8\sqrt{2}},,,\frac{7-\sqrt{7}}{8\sqrt{2}},,,\frac{5-\sqrt{7}}{16\sqrt{2}},,,\frac{1+\sqrt{7}}{16\sqrt{2}}\right],$

or

$h=\left[\frac{-3+\sqrt{15}}{16\sqrt{2}},,,\frac{1-\sqrt{15}}{16\sqrt{2}},,,\frac{3-\sqrt{15}}{8\sqrt{2}},,,\frac{3+\sqrt{15}}{8\sqrt{2}},,,\frac{13+\sqrt{15}}{16\sqrt{2}},,,\frac{9-\sqrt{15}}{16\sqrt{2}}\right],$

or

$h=\left[\frac{-3-\sqrt{15}}{16\sqrt{2}},,,\frac{1+\sqrt{15}}{16\sqrt{2}},,,\frac{3+\sqrt{15}}{8\sqrt{2}},,,\frac{3-\sqrt{15}}{8\sqrt{2}},,,\frac{13-\sqrt{15}}{16\sqrt{2}},,,\frac{9+\sqrt{15}}{16\sqrt{2}}\right],$

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Joseph
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no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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narayan
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ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
How we are making nano material?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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