# 0.5 Approximation of functions

 Page 1 / 1

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 $\stackrel{˜}{N}$ if the primal scaling function $\varphi$ reproduces polynomials up to degree $\stackrel{˜}{N}-1$ , i.e., $\text{For}\phantom{\rule{1.em}{0ex}}0\le p<\stackrel{˜}{N},\phantom{\rule{0.277778em}{0ex}}\exists {c}_{k}\in \mathbb{R}\phantom{\rule{0.277778em}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}{x}^{p}=\sum _{k}{c}_{k}\varphi \left(x-k\right)\phantom{\rule{3.33333pt}{0ex}}.$

The associated dual wavelet $\stackrel{˜}{\psi }$ has then $\stackrel{˜}{N}$ 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 $\stackrel{˜}{N}$ , the higher the regularity of $\varphi$ . Symmetrically to [link] , the order of the dual MRA is $N$ if $\stackrel{˜}{\varphi }$ can reproduce polynomials up to degree $N-1$ . Figure 2 from Multiresolution analysis and wavelets shows an example of a biorthogonal basis where $\stackrel{˜}{N}=3$ and $N=1$ . It illustrates the link between a high number of vanishing moments of thedual wavelet $\stackrel{˜}{\psi }$ and the regularity of the primal scaling function $\varphi$ .

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 ${\stackrel{˜}{\psi }}_{jk}$ have a large number $\stackrel{˜}{N}$ of vanishing moments. Indeed, suppose the function $f$ to be decomposed is analytic on the interval $I$ without discontinuity. Since $〈{x}^{p},\phantom{\rule{0.166667em}{0ex}},,,{\stackrel{˜}{\psi }}_{jk}〉=0$ for $p=0,...,\stackrel{˜}{N}-1$ , we are sure that the first $\stackrel{˜}{N}$ terms of a Taylor expansion of $f$ will not give a contribution to the wavelet coefficient $〈f,\phantom{\rule{0.166667em}{0ex}},,,{\stackrel{˜}{\psi }}_{jk}〉$ provided that the support of ${\stackrel{˜}{\psi }}_{jk}$ 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 $\beta -$ Hölder functions.

## Definition 2

The class ${\Lambda }^{\beta }\left(L\right)$ of Hölder continuous functions is defined as follows:
1. if $\beta \le 1,{\Lambda }^{\beta }\left(L\right)=\left\{f:\left|f,\left(,x,\right),-,f,\left(,y,\right)\right|\le L{|x-y|}^{\beta }\right\}$
2. if $\beta >1,{\Lambda }^{\beta }\left(L\right)=\left\{f,:,\left|{f}^{\left(⌊\beta ⌋\right)},\left(x\right),-,{f}^{\left(⌊\beta ⌋\right)},\left(y\right)\right|,\le ,{L|x-y|}^{{\beta }^{\text{'}}},\phantom{\rule{0.277778em}{0ex}},;,|{f}^{\left(⌊\beta ⌋\right)}\left(x\right)|,\le ,M\right\},$ where $⌊\beta ⌋$ is the largest integer less than $\beta$ and ${\beta }^{\text{'}}=\beta -⌊\beta ⌋.$

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

Let $f\in {\Lambda }^{\beta }\left(L\right)$ , and suppose that the (orthogonal) wavelet function $\psi$ has $r$ continuous derivatives and $r$ vanishing moments with $r>\beta$ . Then
$\left|〈f,\phantom{\rule{0.166667em}{0ex}},,,{\psi }_{jk}〉\right|\le C{2}^{-j\left(\beta +1/2\right)}\phantom{\rule{3.33333pt}{0ex}}.$

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

In the orthogonal setting, the wavelet $\psi$ 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 $\stackrel{˜}{\psi }$ 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 $\stackrel{˜}{\psi }$ may not be very regular. In this case, the wavelet coefficient $〈f,\phantom{\rule{0.166667em}{0ex}},,,{\stackrel{˜}{\psi }}_{jk}〉$ with the less regular wavelet ${\stackrel{˜}{\psi }}_{jk}$ can be used to characterize $f\in {\Lambda }^{\beta }\left(L\right)$ with $0<\beta <\stackrel{˜}{N}$ , even if $\beta >N=1$ : with a biorthogonal basis, regular functions can be characterized by their inner products with much less regular functions.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
Got questions? Join the online conversation and get instant answers!