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The scaling function and the subspaces

There are two ways to introduce wavelets: one is through the continuous wavelet transform, and the other is through multiresolution analysis (MRA), which is the presentation adopted here. Here we start by defining multiresolution analysis and thereafter we give one example of such MRA.

Definition

( Multiresolution analysis) A multiresolution analysis of L 2 ( I R ) is defined as a sequence of closed subspaces V j L 2 ( I R ) , j Z Z with the following properties:
  1. ... V - 1 V 0 V 1 ...
  2. The spaces V j satisfy
    j Z Z V j is dense in L 2 ( I R ) and j Z Z V j = { 0 }
  3. If f ( x ) V 0 , f ( 2 j x ) V j . This property means that all the spaces V j are scaled versions of the central space V 0 .
  4. If f V 0 , f ( . - k ) V 0 , k Z Z . That is, V 0 (and hence all the V j ) is invariant under translation.
  5. There exists ϕ V 0 such that { ϕ 0 , n ; n Z Z } is an orthonormal basis in V 0 .

Condition 5 in [link] seems to be quite contrived, but it can be relaxed (i.e.,instead of taking orthonormal basis, we can take Riesz basis). We will use the following terminology: a level of a multiresolution analysis is one of the V j subspaces and one level is coarser (respectively finer ) with respect to another whenever the index of the corresponding subspace is smaller (respectively bigger).

Consequence of the definition

Let us make a couple of simple observations concerning this definition. Combining the facts that

  1. ϕ ( x ) V 0
  2. { ϕ ( . - k ) , k Z Z } is an orthonormal basis for V 0
  3. ϕ ( 2 j x ) V j ,

we obtain that, for fixed j , { ϕ j , k ( x ) = 2 j / 2 ϕ ( 2 j x - k ) , k Z Z } is an orthonormal basis for V j .

Since ϕ V 0 V 1 , we can express ϕ as a linear combination of { ϕ 1 , k } :

ϕ ( x ) = k h k ϕ 1 , k ( x ) = 2 k h k ϕ ( 2 x - k ) .

[link] is called the refinement equation , or the two scales difference equation. The function ϕ ( x ) is called the scaling function . Under very general condition, ϕ is uniquely defined by its refinement equation and the normalisation

- + ϕ ( x ) d x = 1 .

The spaces V j will be used to approximate general functions (see an example below). This will be done by defining appropriate projections onto these spaces. Since the union of all the V j is dense in L 2 ( I R ) , we are guaranteed that any given function of L 2 can be approximated arbitrarily close by such projections, i.e.:

lim j P j f = f ,

for all f in L 2 . Note that the orthogonal projection of f onto V j can be written as:

P j f = k Z Z α k ϕ j k .

where α k = < f , ϕ j , k > .

Example

The simplest example of a scaling function is given by the Haar function:

ϕ ( x ) = I 1 [ 0 , 1 ] = 1 if 0 x 1 0 otherwise

Hence we have that

ϕ ( 2 x ) = 1 if 0 x 1 / 2 0 otherwise

and

ϕ ( 2 x - 1 ) = 1 if 1 / 2 x 1 0 otherwise

The function ϕ generates, by translation and scaling, a multiresolution analysis for the spaces V j defined by:

V j = { f L 2 ( I R ) ; k Z Z , f | [ 2 j k , 2 j ( k + 1 ) [ = constant }

The wavelet function and the detail spaces wj

The detail space wj

Rather than considering all our nested spaces V j , we would like to code only the information needed to go from V j to V j + 1 . Hence we define by W j the space complementing V j in V j + 1 :

V j + 1 = V j W j

This space W j answers our question: it contains the “detail” information needed to go from an approximation at resolution j to an approximation at resolution j + 1 . Consequently, by using recursively the [link] , we have:

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
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Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Sanket Reply
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Damian Reply
China
Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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Source:  OpenStax, Multiresolution analysis, filterbank implementation, and function approximation using wavelets. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10568/1.2
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