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In this chapter, we give an overview on multiresolution analysis, wavelet series and wavelet estimators in the classical setting. By `classical' or `first-generation' wavelets, we mean waveletsthat were constructed initially to analyze signals observed at equispaced design points and having a sample size which is a power of two.The `second-generation' wavelet basis presented in the subsequent chapters will release these two constraints.

If one wants to analyze a function of time with a series expansion, the first idea that comes probably into one's mind is to use a Fourier series, i.e. decompose the function into sine and cosine at different frequencies. In this process, we hope that only a few coefficients in the series will carry mostof the information about the signal. Certain smooth functions have such an `economical' Fourier expansion. However, for most functions, a good Fourier seriesapproximation requires numerous sine and cosine basis functions. Indeed, the sine functions have a precise frequency but are not localized in time, hence a localizedinformation in the signal like a discontinuity will affect all the coefficients of the series. This drawback lead the researchers to look for more efficient bases, that is,bases which are localized both in time and in frequency. We will see in Multiresolution analysis and wavelets that a wavelet basis offers this property.

This chapter is structured as follows. We begin by recalling some notations in Function spaces: notion and notations . Next Multiresolution analysis and wavelets introduces the multiresolution analysis, the wavelet functions, and gives some simple examples of wavelet bases. Fast wavelet transform explains how to decompose a signal using the wavelet transform. Such wavelet transforms, also called `decimated', lack the property of translation-invariance. Non-decimated wavelet transform presents a widely used trick to make a wavelet transform translation-invariant. Since the main goal of a wavelet series is to providea good approximation of a function belonging to a given space, Approximation of Functions introduces some fundamental notions to measure the quality of such approximation. Finally, Nonparametric regression with wavelets presents how to construct a nonparametric regression estimator using wavelets. First, the classical case of equally spaced design is considered. In the last part of Nonparametric regression with wavelets , we review some existing methods that deal with irregular designs.

Questions & Answers

what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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Source:  OpenStax, An introduction to wavelet analysis. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10566/1.3
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