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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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