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We first give a definition of the order of a multiresolution analysis.

(Order of a MRA in the classical setting) A multiresolution analysis is said to be of order N ˜ if the primal scaling function ϕ reproduces polynomials up to degree N ˜ - 1 , i.e., For 0 p < N ˜ , c k R such that x p = k c k ϕ ( x - k ) .

The associated dual wavelet ψ ˜ has then N ˜ vanishing moments.In the classical setting, it is proved that the order of a MRA and the regularity of the scaling function are linked: the larger N ˜ , the higher the regularity of ϕ . Symmetrically to [link] , the order of the dual MRA is N if ϕ ˜ can reproduce polynomials up to degree N - 1 . Figure 2 from Multiresolution analysis and wavelets shows an example of a biorthogonal basis where N ˜ = 3 and N = 1 . It illustrates the link between a high number of vanishing moments of thedual wavelet ψ ˜ and the regularity of the primal scaling function ϕ .

The main objective when decomposing a function in a wavelet series is to create a sparse representation of the function, that is, to obtain a decomposition where only a few number of detail coefficients are `large', while the majority of the coefficients are close to zero. By `large', we mean that the absolute value of the detail coefficient is large.

Near a singularity, large detail coefficients at different levels will be needed to recover the discontinuity. However, between points of singularity, we can hope to have small detail coefficients, in particular if the analyzing wavelets ψ ˜ j k have a large number N ˜ of vanishing moments. Indeed, suppose the function f to be decomposed is analytic on the interval I without discontinuity. Since x p , ψ ˜ j k = 0 for p = 0 , ... , N ˜ - 1 , we are sure that the first N ˜ terms of a Taylor expansion of f will not give a contribution to the wavelet coefficient f , ψ ˜ j k provided that the support of ψ ˜ j k does not contain any singularities of the function f .

This sparse representation explains why classical wavelets providesmoothness characterization of function spaces like the Hölder and Sobolev spaces [link] , but also of more general Besov spaces, which may contain functions of inhomogeneousregularity [link] , [link] , [link] , [link] , [link] .

We illustrate this characterization property with the case of β - Hölder functions.

Definition 2

The class Λ β ( L ) of Hölder continuous functions is defined as follows:
  1. if β 1 , Λ β ( L ) = f : f ( x ) - f ( y ) L | x - y | β
  2. if β > 1 , Λ β ( L ) = f : f β ( x ) - f β ( y ) L | x - y | β ' ; | f β ( x ) | M , where β is the largest integer less than β and β ' = β - β .

The global Hölder regularity of a function can be characterized as follows [link] , [link] .

Let f Λ β ( L ) , and suppose that the (orthogonal) wavelet function ψ has r continuous derivatives and r vanishing moments with r > β . Then
f , ψ j k C 2 - j ( β + 1 / 2 ) .

A similar characterization exists for continuous and Sobolev functions [link] , [link] .

In the orthogonal setting, the wavelet ψ must be regular and have a high number of vanishing moments. On the contrary,in the biorthogonal expansion equation 5 from Multiresolution analysis and wavelets , it is mostly of interest to have a dual wavelet ψ ˜ with a high number of vanishing moments, and hence a regular primal scaling and wavelet functions. On the primal side, it is sufficient to have only one vanishing moment for wavelet denoising, and consequently ψ ˜ may not be very regular. In this case, the wavelet coefficient f , ψ ˜ j k with the less regular wavelet ψ ˜ j k can be used to characterize f Λ β ( L ) with 0 < β < N ˜ , even if β > N = 1 : with a biorthogonal basis, regular functions can be characterized by their inner products with much less regular functions.

Questions & Answers

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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Source:  OpenStax, An introduction to wavelet analysis. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10566/1.3
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