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Given a mother scaling function φ t 2 —the choice of which will be discussed later—let us construct scaling functions at"coarseness-level- k" and "shift- n " as follows: φ k , n t 2 k 2 φ 2 k t n . Let us then use V k to denote the subspace defined by linear combinations of scaling functions at the k th level: V k span φ k , n t n . In the Haar system, for example, V 0 and V 1 consist of signals with the characteristics of x 0 t and x 1 t illustrated in .

We will be careful to choose a scaling function φ t which ensures that the following nesting property is satisfied: V 2 V 1 V 0 V -1 V -2 coarse detailed In other words, any signal in V k can be constructed as a linear combination of more detailed signals in V k 1 . (The Haar system gives proof that at least one such φ t exists.)

The nesting property can be depicted using the set-theoretic diagram, , where V 1 is represented by the contents of the largest egg (which includes the smaller two eggs), V 0 is represented by the contents of the medium-sized egg (which includes the smallest egg), and V 1 is represented by the contents of the smallest egg.

Going further, we will assume that φ t is designed to yield the following three important properties:

  • φ k , n t n constitutes an orthonormal basis for V k ,
  • V 0 (contains no signals).
    While at first glance it might seem that V should contain non-zero constant signals ( e.g. , x t a for a ), the only constant signal in 2 , the space of square-integrable signals, is the zero signal.
  • V 2 (contains all signals).
Because φ k , n t n is an orthonormal basis, the best (in 2 norm) approximation of x t 2 at coarseness-level- k is given by the orthogonal projection,
x k t n c k , n φ k , n t
c k , n φ k , n t x t

We will soon derive conditions on the scaling function φ t which ensure that the properties above are satisfied.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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