# 0.4 Characterization by approximation properties

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The following is a short introduction to Besov spaces and their characterization by means of approximationprocedures as well as wavelet decompositions.

An important feature of Besov spaces is that they admit equivalent characterization by multiresolution approximation properties and by wavelet decompositions.

Here we use the following standard notation (see [link] or [link] for a general treatment): if $f$ is function we denote by ${P}_{j}f$ its projection onto the space ${V}_{j}$ , and by ${Q}_{j}f={P}_{j+1}f-{P}_{j}f$ its projection onto the detail space ${W}_{j}$ . The multiscale decomposition of $f$ writes

$f={P}_{0}f+\sum _{j\ge 0}{Q}_{j}f.$

The projectors ${P}_{j}$ and ${Q}_{j}$ can be further expressed in terms of biorthogonal scaling functions and wavelets bases:

${P}_{j}f:=\sum _{|\lambda |=j}⟨f,{\stackrel{˜}{\varphi }}_{\lambda }⟩{\varphi }_{\lambda }\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{j}f:=\sum _{|\lambda |=j}⟨f,{\stackrel{˜}{\psi }}_{\lambda }⟩{\psi }_{\lambda }.$

Here we use the simplified notation ${\varphi }_{\lambda }$ with “ $|\lambda |=j$ ” meaning that the functions are picked at resolution $j$ . In the case where $\Omega ={\mathbb{R}}^{d}$ , thesehave the general from ${\varphi }_{\lambda }\left(x\right):={\varphi }_{j,k}\left(x\right):={2}^{dj/2}\varphi \left({2}^{j}x-k\right)$ , bur for a general domain $\Omega ={\mathbb{R}}^{d}$ proper adaptations of these bases need to be done near the boundary. We can thereforewrite

$f=\sum {d}_{\lambda }{\psi }_{\lambda },\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{d}_{\lambda }:=⟨f,{\stackrel{˜}{\psi }}_{\lambda }⟩,$

where we include in this sum the wavelets at all levels $j\ge 0$ and we incorporate the scaling function ${\varphi }_{\lambda }$ at the first level $|\lambda |=0$ .

Under certain assumptions that we shall discuss below, it is known that the Besov norm ${\parallel f\parallel }_{{B}_{p,q}^{s}}$ is equivalent to

$\parallel {P}_{0}{f\parallel }_{{L}^{p}}+\parallel \left({2}^{sj}\parallel f-{P}_{j}{f\parallel }_{{L}^{p}}{{\right)}_{j\ge 0}\parallel }_{{\ell }^{q}},$

or to

$\parallel {P}_{0}{f\parallel }_{{L}^{p}}+\parallel \left({2}^{sj}\parallel {Q}_{j}{f\parallel }_{{L}^{p}}{{\right)}_{j\ge 0}\parallel }_{{\ell }^{q}}.$

Using the equivalence $\parallel {Q}_{j}{f\parallel }_{{L}^{p}}\sim {2}^{\left(d/2-d/p\right)j}{\parallel {\left({d}_{\lambda }\right)}_{|\lambda |=j}\parallel }_{{\ell }^{p}}$ at each level to prove a third equivalent norm interms of the wavelet coefficients:

$\parallel \left({2}^{sj}{2}^{\left(d/2-d/p\right)j}\parallel {\left({d}_{\lambda }\right)}_{|\lambda |=j}{\parallel }_{{\ell }^{p}}{{\right)}_{j\ge 0}\parallel }_{{\ell }^{q}}.$

These equivalences mean that the modulus of smoothness ${\omega }_{n}{\left(f,{2}^{-j}\right)}_{{L}^{p}}$ in the definition of ${B}_{p,q}^{s}$ can be replaced either by $\parallel f-{P}_{j}{f\parallel }_{{L}^{p}}$ or by $\parallel {Q}_{j}{f\parallel }_{{L}^{p}}$ . Their validity requires thatthe spaces ${V}_{j}$ satisfy the following two assumptions:

• The ${V}_{j}$ must satisfy an approximation property that takes the form of a direct estimate
$\parallel f-{P}_{j}{f\parallel }_{{L}^{p}}\le C{\omega }_{n}{\left(f,{2}^{-j}\right)}_{{L}^{p}}.$
Such an estimate ensures that a smooth function will have a fast rate of approximation.
• They must also satisfy smoothness properties that takes the form of an inverse estimate
${\omega }_{n}{\left({f}_{j},t\right)}_{{L}^{p}}\le C{\left[min\left(1,t{2}^{j}\right)\right]}^{n}{\parallel {f}_{j}\parallel }_{{L}^{p}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{f}_{j}\in {V}_{j}.$
Such an estimate takes into account the smoothness of the spaces ${V}_{j}$ : it ensuresthat a function that is approximated at a sufficiently fast rate rate by these spacesshould also have some smoothness.

One can show that the direct estimate is satisfied if and only if all polynomials up to order $n-1$ can be written as combinations of the scaling functions ${\varphi }_{\lambda }$ in ${V}_{j}$ , or equivalently if the dual wavelets ${\stackrel{˜}{\psi }}_{\lambda }$ have $n$ vanishing moments. On the other hand, the inverse estimate requires that the scaling functions ${\varphi }_{\lambda }$ that generates ${V}_{j}$ are smooth in the sense of belonging to ${W}^{n,p}$ . Note that the direct estimate immediately implies that theexpression [link] is less than ${\parallel f\parallel }_{{B}_{p,q}^{s}}$ . A more refined mechanism, using theinverse estimate (as well as some discrete Hardy inequalities) is used to prove the full equivalence between ${\parallel f\parallel }_{{B}_{p,q}^{s}}$ and [link] or [link] . We refer to chapter III in [link] for a detailed proof of these results.

These equivalences show that the convergence rate ${N}^{-t/d}$ ( $N=\mathrm{dim}\left({V}_{j}\right)$ ) can be achieved by the linearmultiscale approximation process $f↦{P}_{f}$ , if and only if the function has roughly “ $t$ derivatives in ${L}^{p}$ ”.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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