# 0.5 The scaling function and scaling coefficients, wavelet and

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We will now look more closely at the basic scaling function and wavelet to see when they exist and what their properties are [link] , [link] , [link] , [link] , [link] , [link] , [link] . Using the same approach that is used in the theory of differentialequations, we will examine the properties of $\phi \left(t\right)$ by considering the equation of which it is a solution. The basic recursion [link] that comes from the multiresolution formulation is

$\phi \left(t\right)=\sum _{n}h\left(n\right)\phantom{\rule{0.277778em}{0ex}}\sqrt{2}\phantom{\rule{0.166667em}{0ex}}\phi \left(2t-n\right)$

with $h\left(n\right)$ being the scaling coefficients and $\phi \left(t\right)$ being the scaling function which satisfies this equation which is sometimes calledthe refinement equation , the dilation equation , or the multiresolution analysis equation (MRA).

In order to state the properties accurately, some care has to be taken in specifying just what classes of functions are being considered or areallowed. We will attempt to walk a fine line to present enough detail to be correct but not so much as to obscure the main ideas and results. A fewof these ideas were presented in Section: Signal Spaces and a few more will be given in the next section. A more complete discussion can be foundin [link] , in the introductions to [link] , [link] , [link] , or in any book on function analysis.

## Signal classes

There are three classes of signals that we will be using. The most basic is called ${L}^{2}\left(\mathbf{R}\right)$ which contains all functions which have a finite, well-defined integral of the square: $f\in {L}^{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}⇒\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\int {|f\left(t\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}dt=E<\infty$ . This class is important because it is a generalization of normal Euclideangeometry and because it gives a simple representation of the energy in a signal.

The next most basic class is ${L}^{1}\left(\mathbf{R}\right)$ , which requires a finite integral of the absolute value of the function: $f\in {L}^{1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}⇒\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\int |f\left(t\right)|\phantom{\rule{0.166667em}{0ex}}dt=K<\infty$ . This class is important because one may interchange infinite summations and integrations with these functions although not necessarily with ${L}^{2}$ functions. These classes of function spaces can be generalized to those with ${\int |f\left(t\right)|}^{p}\phantom{\rule{0.166667em}{0ex}}dt=K<\infty$ and designated ${L}^{p}$ .

A more general class of signals than any ${L}^{p}$ space contains what are called distributions . These are generalized functions which are not defined by their having “values" but by the value of an “inner product"with a normal function. An example of a distribution would be the Dirac delta function $\delta \left(t\right)$ where it is defined by the property: $f\left(T\right)=\int f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(t-T\right)\phantom{\rule{0.166667em}{0ex}}dt$ .

Another detail to keep in mind is that the integrals used in these definitions are Lebesque integrals which are somewhat more general than the basic Riemann integral. The value of a Lebesque integral is notaffected by values of the function over any countable set of values of its argument (or, more generally, a set of measure zero). A function definedas one on the rationals and zero on the irrationals would have a zero Lebesque integral. As a result of this, properties derived using measuretheory and Lebesque integrals are sometime said to be true “almost everywhere," meaning they may not be true over a set of measure zero.

Many of these ideas of function spaces, distributions, Lebesque measure, etc. came out of the early study of Fourier series and transforms. It isinteresting that they are also important in the theory of wavelets. As with Fourier theory, one can often ignore the signal space classes and canuse distributions as if they were functions, but there are some cases where these ideas are crucial. For an introductory reading of this bookor of the literature, one can usually skip over the signal space designation or assume Riemann integrals. However, when a contradiction orparadox seems to arise, its resolution will probably require these details.

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?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
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
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
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?
how did you get the value of 2000N.What calculations are needed to arrive at it
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