# 0.4 Bases, orthogonal bases, biorthogonal bases, frames, tight

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Development of ideas of vector expansion

Most people with technical backgrounds are familiar with the ideas of expansion vectors or basis vectors and of orthogonality; however, therelated concepts of biorthogonality or of frames and tight frames are less familiar but also important. In the study of wavelet systems, we find thatframes and tight frames are needed and should be understood, at least at a superficial level. One can find details in [link] , [link] , [link] , [link] , [link] . Another perhaps unfamiliar concept is that of an unconditional basis usedby Donoho, Daubechies, and others [link] , [link] , [link] to explain why wavelets are good for signal compression, detection, and denoising [link] , [link] . In this chapter, we will very briefly define and discuss these ideas. At this point, you may want to skip thesesections and perhaps refer to them later when they are specifically needed.

## Bases, orthogonal bases, and biorthogonal bases

A set of vectors or functions ${f}_{k}\left(t\right)$ spans a vector space $F$ (or $F$ is the Span of the set) if any element of that space can be expressed as a linear combination of members of thatset, meaning: Given the finite or infinite set of functions ${f}_{k}\left(t\right)$ , we define ${\mathrm{Span}}_{k}\left\{{f}_{k}\right\}=F$ as the vector space with all elements of the space of the form

$g\left(t\right)=\sum _{k}\phantom{\rule{0.277778em}{0ex}}{a}_{k}\phantom{\rule{0.277778em}{0ex}}{f}_{k}\left(t\right)$

with $k\in \mathbf{Z}$ and $t,a\in \mathbf{R}$ . An inner product is usually defined for this space and is denoted $⟨f\left(t\right),g\left(t\right)⟩$ . A norm is defined and is denoted by $\parallel f\parallel =\sqrt{⟨f,f⟩}$ .

We say that the set ${f}_{k}\left(t\right)$ is a basis set or a basis for a given space $F$ if the set of $\left\{{a}_{k}\right\}$ in [link] are unique for any particular $g\left(t\right)\in F$ . The set is called an orthogonal basis if $⟨{f}_{k}\left(t\right),{f}_{\ell }\left(t\right)⟩=0$ for all $k\ne \ell$ . If we are in three dimensional Euclidean space, orthogonal basis vectors are coordinate vectors that are at right (90 o ) angles to each other. We say the set is an orthonormal basis if $⟨{f}_{k}\left(t\right),{f}_{\ell }\left(t\right)⟩=\delta \left(k-\ell \right)$ i.e. if, in addition to being orthogonal, the basis vectors are normalized to unity norm: $\parallel {f}_{k}\left(t\right)\parallel =1$ for all $k$ .

From these definitions it is clear that if we have an orthonormal basis, we can express any element in the vector space, $g\left(t\right)\in F$ , written as [link] by

$g\left(t\right)=\sum _{k}⟨g\left(t\right),\phantom{\rule{0.166667em}{0ex}}{f}_{k}\left(t\right)⟩\phantom{\rule{0.277778em}{0ex}}{f}_{k}\left(t\right)$

since by taking the inner product of ${f}_{k}\left(t\right)$ with both sides of [link] , we get

${a}_{k}=⟨g\left(t\right),\phantom{\rule{0.166667em}{0ex}}{f}_{k}\left(t\right)⟩$

where this inner product of the signal $g\left(t\right)$ with the basis vector ${f}_{k}\left(t\right)$ “picks out" the corresponding coefficient ${a}_{k}$ . This expansion formulation or representation is extremely valuable. It expresses [link] as an identity operator in the sense that the inner product operates on $g\left(t\right)$ to produce a set of coefficients that, when used to linearly combine the basis vectors, gives back the original signal $g\left(t\right)$ . It is the foundation of Parseval's theorem which says the norm or energycan be partitioned in terms of the expansion coefficients ${a}_{k}$ . It is why the interpretation, storage, transmission, approximation, compression, andmanipulation of the coefficients can be very useful. Indeed, [link] is the form of all Fourier type methods.

Although the advantages of an orthonormal basis are clear, there are cases where the basis system dictated by the problem is not and cannot (orshould not) be made orthogonal. For these cases, one can still have the expression of [link] and one similar to [link] by using a dual basis set ${\stackrel{˜}{f}}_{k}\left(t\right)$ whose elements are not orthogonal to each other, but to the corresponding element of the expansion set

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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
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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