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Folding Functions Supported on Half-Lines.
Folding Functions Supported on Half-Lines:
(a) f ( t ) = cos ( π 2 11 t ) u ( t ) ( ( u ( t ) is the Unit Step or Heaviside Function)(b) U + ( r [ 3 ] , 0 , 1 ) f ( t )
(c) f ( t ) = sin ( π 2 11 t ) u ( - t )
(d) U + ( r [ 3 ] , 0 , 1 ) f ( t )

Local cosine and sine bases

Recall the four orthonormal trigonometric bases for L 2 ( ( 0 , 1 ) ) we described earlier.

  1. Φ n ( t ) = 2 cos ( π ( n + 1 2 ) t ) , n 0 , 1 , 2 , ... ;
  2. Φ n ( t ) = 2 sin ( π ( n + 1 2 ) t ) , n 0 , 1 , 2 , ... ;
  3. Φ n ( t ) = 1 , 2 cos ( π n t ) , n 1 , 2 , ... ;
  4. Φ n ( t ) = 2 sin ( π n t ) , n 0 , 1 , 2 , ... .

The bases functions have discontinuities at t = 0 and t = 1 because they are restrictions of the cosines and sines to the unit interval by rectangular windowing.The natural extensions of these basis functions to t (i.e., unwindowed cosines and sines) are either even (say “+”) or odd (say “-”)symmetric (locally) about the endpoints t = 0 and t = 1 . Indeed the basis functions for the four cases are ( + , - ) , ( - , + ) , ( + , + ) and ( - , - ) symmetric, respectively, at ( 0 , 1 ) . From the preceding analysis, this means that unfolding these basis functionscorresponds to windowing if the unfolding operator has the right polarity. Also observe that the basis functions are discontinuous atthe endpoints. Moreover, depending on the symmetry at each endpoint all odd derivatives (for “+” symmetry) or evenderivatives (for “ - ” symmetry) are zero. By choosing unfolding operators of appropriate polarity at the endpoints (with non overlapping actionregions) for the four bases, we get smooth basis functions of compact support. For example, for (+, - ) symmetry, the basis function U + ( r 0 , 0 , ϵ 0 ) U + ( r 1 , 1 , ϵ 1 ) ψ n ( t ) is supported in ( - ϵ 0 , 1 + ϵ 1 ) and is as many times continuously differentiable as r 0 and r 1 are.

Let t j be an ordered set of points in defining a partition into disjoint intervals I j = [ t j , t j + 1 ] . Now choose one of the four bases above for each intervalsuch that at t j the basis functions for I j - 1 and that for I j have opposite symmetries. We say the polarity at t j is positive if the symmetry is - ) ( + and negative if it is + ) ( - . At each t j choose a smooth cutoff function r j ( t ) and action radius ϵ j so that the action intervals do not overlap. Let p ( j ) be the polarity of t j and define the unitary operator

U = j U p ( j ) ( r j , t j , ϵ j ) .

Let ψ n ( t ) denote all the basis functions for all the intervals put together. Then ψ n ( t ) forms a nonsmooth orthonormal basis for L 2 ( ) . Simultaneously U ψ n ( t ) also forms a smooth

orthonormal basis for L 2 ( ) . To find the expansion coefficients of a function f ( t ) in this basis we use

f , U ψ n = U f , ψ n .

In other words, to compute the expansion coefficients of f in the new (smooth) basis, one merely folds f to U f and finds its expansion coefficients with respect to the originalbasis. This allows one to exploit fast algorithms available for coefficient computation in the original basis.

So for an arbitrary choice of polarities at the end points t j we have smooth local trigonometric bases. In particular by choosing the polarityto be positive for all t j (consistent with the choice of the first basis in all intervals) we get local cosine bases.If the polarity is negative for all t j (consistent with the choice of the second basis for all intervals), we get localsine bases. Alternating choice of polarity (consistent with the alternating choice of the third and fourth bases in the intervals) thusleads to alternating cosine/sine bases.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
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
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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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