<< Chapter < Page Chapter >> Page >

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 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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, An introduction to wavelet analysis. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10566/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'An introduction to wavelet analysis' conversation and receive update notifications?