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g ( t ) = k a k cos ( k t )

where the basis vectors (functions) are

f k ( t ) = cos ( k t )

and the expansion coefficients are obtained as

a k = g ( t ) , f k ( t ) = 2 π 0 π g ( t ) cos ( k t ) d x .

The basis vector set is easily seen to be orthonormal by verifying

f ( t ) , f k ( t ) = δ ( k - ) .

These basis functions span an infinite dimensional vector space and the convergence of [link] must be examined. Indeed, it is the robustness of that convergence that is discussed in this section under the topic ofunconditional bases.

Sinc expansion example

Another example of an infinite dimensional orthogonal basis is Shannon's sampling expansion [link] . If f ( t ) is band limited, then

f ( t ) = k f ( T k ) sin ( π T t - π k ) π T t - π k

for a sampling interval T < π W if the spectrum of f ( t ) is zero for | ω | > W . In this case the basis functions are the sinc functions with coefficients which are simply samples of the originalfunction. This means the inner product of a sinc basis function with a bandlimited function will give a sample of that function. It is easy tosee that the sinc basis functions are orthogonal by taking the inner product of two sinc functions which will sample one of them at the pointsof value one or zero.

Frames and tight frames

While the conditions for a set of functions being an orthonormal basis are sufficient for the representation in [link] and the requirement of the set being a basis is sufficient for [link] , they are not necessary. To be a basis requires uniqueness of the coefficients. Inother words it requires that the set be independent , meaning no element can be written as a linear combination of the others.

If the set of functions or vectors is dependent and yet does allow the expansion described in [link] , then the set is called a frame [link] . Thus, a frame is a spanning set . The term frame comes from a definition that requires finite limits on an inequality bound [link] , [link] of inner products.

If we want the coefficients in an expansion of a signal to represent the signal well, these coefficients should have certain properties. They arestated best in terms of energy and energy bounds. For an orthogonal basis, this takes the form of Parseval's theorem. To be a frame in asignal space, an expansion set ϕ k ( t ) must satisfy

A g 2 k | φ k , g | 2 B g 2

for some 0 < A and B < and for all signals g ( t ) in the space. Dividing [link] by g 2 shows that A and B are bounds on the normalized energy of the inner products. They “frame" thenormalized coefficient energy. If

A = B

then the expansion set is called a tight frame . This case gives

A g 2 = k | φ k , g | 2

which is a generalized Parseval's theorem for tight frames. If A = B = 1 , the tight frame becomes an orthogonal basis. From this, it can be shown thatfor a tight frame [link]

g ( t ) = A - 1 k φ k ( t ) , g ( t ) φ k ( t )

which is the same as the expansion using an orthonormal basis except for the A - 1 term which is a measure of the redundancy in the expansion set.

If an expansion set is a non tight frame, there is no strict Parseval's theorem and the energy in the transform domain cannot be exactly partitioned. However,the closer A and B are, the better an approximate partitioning can be done. If A = B , we have a tight frame and the partitioning can be done exactly with [link] . Daubechies [link] shows that the tighter the frame bounds in [link] are, the better the analysis and synthesis system is conditioned. In other words, if A is near or zero and/or B is very large compared to A , there will be numerical problems in the analysis–synthesis calculations.

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
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.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
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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
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Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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