# 0.4 Bases, orthogonal bases, biorthogonal bases, frames, tight  (Page 3/5)

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$g\left(t\right)=\sum _{k}\phantom{\rule{0.166667em}{0ex}}{a}_{k}\phantom{\rule{0.277778em}{0ex}}cos\left(kt\right)$

where the basis vectors (functions) are

${f}_{k}\left(t\right)=cos\left(kt\right)$

and the expansion coefficients are obtained as

${a}_{k}=⟨g\left(t\right),\phantom{\rule{0.166667em}{0ex}}{f}_{k}\left(t\right)⟩=\frac{2}{\pi }{\int }_{0}^{\pi }g\left(t\right)\phantom{\rule{0.166667em}{0ex}}cos\left(kt\right)\phantom{\rule{0.166667em}{0ex}}dx.$

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

$⟨{f}_{\ell }\left(t\right),\phantom{\rule{0.166667em}{0ex}}{f}_{k}\left(t\right)⟩=\delta \left(k-\ell \right).$

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 $f\left(t\right)$ is band limited, then

$f\left(t\right)\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\sum _{k}f\left(Tk\right)\phantom{\rule{0.166667em}{0ex}}\frac{sin\left(\frac{\pi }{T}t-\pi k\right)}{\frac{\pi }{T}t-\pi k}$

for a sampling interval $T<\frac{\pi }{W}$ if the spectrum of $f\left(t\right)$ is zero for $|\omega |>W$ . 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 ${\varphi }_{k}\left(t\right)$ must satisfy

${A\parallel g\parallel }^{2}\le \sum _{k}|⟨{\phi }_{k},g⟩{|}^{2}\le B{\parallel g\parallel }^{2}$

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

$A\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}B$

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

${A\parallel g\parallel }^{2}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\sum _{k}{|⟨{\phi }_{k},g⟩|}^{2}$

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]

$g\left(t\right)={A}^{-1}\sum _{k}⟨{\phi }_{k}\left(t\right),g\left(t\right)⟩\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\phi }_{k}\left(t\right)$

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.

where we get a research paper on Nano chemistry....?
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
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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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
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Alexandre
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Rafiq
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Damian
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What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
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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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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