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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

who was the first nanotechnologist
Lizzy Reply
technologist's thinker father is Richard Feynman but the literature first user scientist Nario Tagunichi.
Norio Taniguchi
I need help
anyone have book of Abdel Salam Hamdy Makhlouf book in pdf Fundamentals of Nanoparticles: Classifications, Synthesis
Naeem Reply
what happen with The nano material on The deep space.?
pedro Reply
It could change the whole space science.
the characteristics of nano materials can be studied by solving which equation?
sibaram Reply
plz answer fast
synthesis of nano materials by chemical reaction taking place in aqueous solvents under high temperature and pressure is call?
hydrothermal synthesis
how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
STM - Scanning Tunneling Microscope.
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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