# 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.

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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NANO
how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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