# 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

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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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