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j Z Z W j = L 2 ( I R ) .

The main interest of MRA lies in the fact that, whenever a collection of closed subspaces satisfies the conditions in [link] , there exists an orthonormal wavelet basis, noted { ψ j , k , k Z Z } of W j . Hence, we can say in very general terms that the projection of a function f onto V j + 1 can be decomposed as

P j + 1 f = P j f + Q j f ,

where P j is the projection operator onto V j , and Q j is the projection operator onto W j . Before proceeding further, let us state clearly what the term “orthogonal wavelet ” involves.

Recapitulation: orthogonal wavelet

We introduce above the class of orthogonal wavelets. Let us define this precisely.

An orthogonal wavelet basis is associated with an orthogonal multiresolution analysis that can be defined as follows. We talk about orthogonal MRA when the wavelet spaces W j are defined as the orthogonal complement of V j in V j + 1 . Consequently, the spaces W j , with j Z Z are all mutually orthogonal, the projections P j and Q j are orthogonal, so that the expansion

f ( x ) = j Q j f ( x )

is orthogonal. Moreover, as Q j is orthogonal, the projection onto W j can explicitely be written as:

Q j f ( x ) = k β j k ψ j k ( x ) ,


β j k = < f , ψ j k > .

A sufficient condition for a MRA to be orthogonal is:

W 0 V 0 ,

or < ψ , ϕ ( . - l ) > = 0 for l Z Z , since the other conditions simply follow from scaling.

An orthogonal wavelet is a function ψ such that the collection of functions { ψ ( x - l ) | l Z Z } is an orthonormal basis of W 0 . This is the case if < ψ , ψ ( . - l ) > = δ l , 0 .

In the following, we consider only orthogonal wavelets. We now outline how to construct a wavelet function ψ ( x ) starting from ϕ ( x ) , and thereafter we show what this construction gives with the Haar function.

How to construct ψ(χ) starting from ρ(χ)?

Suppose we have an orthonormal basis (ONB) { ϕ j , k , k Z Z } for V j and we want to construct ψ j k such that

  • ψ j k , k Z Z form an ONB for W j
  • V j W j , i.e. < ϕ j k , ψ j k ' > = 0 k , k '
  • W j W j ' for j j ' .

It is natural to use conditions given by the MRA aspect to obtain this. More specifically, the following relationships are used to characterize ψ :

  1. Since ϕ V 0 V 1 , and the ϕ 1 , k are an ONB in V 1 , we have:
    ϕ ( x ) = k h k ϕ 1 , k , h k = < ϕ , ϕ 1 , k > , k Z Z | h k | 2 = 1 .
    (refinement equation)
  2. δ k , 0 = ϕ ( x ) ϕ ( x - k ) ¯ d x
    (orthonormality of ϕ ( . - k ) )
  3. Let us now characterize W 0 : f W 0 is equivalent to f V 1 and f V 0 . Since f V 1 , we have:
    f = n f n ϕ 1 , n , with f n = < f , ϕ 1 , n > .
    The constraint f V 0 is implied by f ϕ 0 , k for all k .
  4. Taking the general form of f W 0 , we can deduce a candidate for our wavelet. We then need to verify that the ψ 0 , k are indeed an ONB of W 0 .

In fact, in our setting, the conditions given above can be re-written in the Fourier domain, where the manipulations become easier (for details,see [link] , chapter 5). Let us now state the result of these manipulations.

(Daubechies, chap 5) If a ladder of closed subspaces ( V j ) j Z Z in L 2 ( I R ) satisfies the conditions of the [link] , then there exists an associated orthonormal wavelet basis { ψ j , k ; j , k Z Z } for L 2 ( I R ) such that:
P j + 1 = P j + k < . , ψ j , k > ψ j , k .

One possibility for the construction of the wavelet ψ is to take

ψ ( x ) = ( 2 ) n ( - 1 ) n h 1 - n ϕ ( 2 x - n )

(with convergence of this serie is L 2 sense).


Let us see what the recipe of [link] gives for the Haar multiresolution analysis. In that case, ϕ ( x ) = 1 for 0 x 1 , 0 otherwise. Hence:

h n = 2 d x ϕ ( x ) ϕ ( 2 x - n ) = 1 / 2 if n = 0 , 1 0 otherwise


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

Homogeneous and inhomogeneous representation of

Inhomogeneous representation

If we consider a first (coarse) approximation of f V 0 , and then “refine” this approximation with detail spaces W j , the decomposition of f can be written as:

f = P 0 f + j = 0 Q j f = k α k ϕ 0 , k ( x ) + j = 0 k β j , k ψ j , k ( x ) ,

where α k = < f , ϕ 0 , k > and β j , k = < f , ψ j , k > . In this case, we talk about inhomogeneous representation of f .

Homogeneous representation

If we use the fact that

j Z Z W j is dense in L 2 ( I R ) ,

we can decompose f as a linear combination of functions ψ j , k only:

f ( x ) = j = - + k β j , k ψ j , k ( x ) .

We then talk about homogeneous representation of f .

Properties of the homogeneous representation

  • Each coefficient β j , k in [link] depends only locally on f because
    β j , k = f ( x ) ψ j , k ( x ) d x ,
    and the wavelet ψ j , k ( x ) has (essentially) bounded support.
  • β j , k gives information on scale 2 - j , near position 2 - j k
  • a discontinuity in f only affects a small proportion of coefficients– a fixed number at each frequency level.

Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
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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.
Do you know which machine is used to that process?
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
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what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
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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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