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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 ( t ) 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 ( t ) , we define Span k { f k } = F as the vector space with all elements of the space of the form

g ( t ) = k a k f k ( t )

with k Z and t , a R . An inner product is usually defined for this space and is denoted f ( t ) , g ( t ) . A norm is defined and is denoted by f = f , f .

We say that the set f k ( t ) is a basis set or a basis for a given space F if the set of { a k } in [link] are unique for any particular g ( t ) F . The set is called an orthogonal basis if f k ( t ) , f ( t ) = 0 for all k . 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 ( t ) , f ( t ) = δ ( k - ) i.e. if, in addition to being orthogonal, the basis vectors are normalized to unity norm: f k ( t ) = 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 ( t ) F , written as [link] by

g ( t ) = k g ( t ) , f k ( t ) f k ( t )

since by taking the inner product of f k ( t ) with both sides of [link] , we get

a k = g ( t ) , f k ( t )

where this inner product of the signal g ( t ) with the basis vector f k ( t ) “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 ( t ) to produce a set of coefficients that, when used to linearly combine the basis vectors, gives back the original signal g ( t ) . 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 f ˜ k ( t ) whose elements are not orthogonal to each other, but to the corresponding element of the expansion set

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
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I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
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I rally confuse this number And equations too I need exactly help
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Commplementary angles
Idrissa Reply
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
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
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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