# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 22/28)

 Page 22 / 28

## Nonsmooth local trigonometric bases

To construct local trigonometric bases we have to choose: (a) the window functions ${w}_{k}\left(t\right)$ ; and (b) the trigonometric functions (i.e., $\alpha$ , $\beta$ and $\gamma$ in Eq.  [link] ). If we use the rectangular window (which we know is a bad choice),then it suffices to find a trigonometric basis for the interval that the window spans. Without loss of generality, we couldconsider the unit interval $\left(0,1\right)$ and hence we are interested in trigonometric bases for ${L}^{2}\left(\left(0,1\right)\right)$ . It is easy to see that the following four sets of functions satisfy this requirement.

1. $\left\{{\Phi }_{n},\left(t\right)\right\}=\left\{\sqrt{2},cos,\left(\pi \left(n+\frac{1}{2}\right)t\right)\right\}$ , $n\in \left\{0,,,1,,,2,,,...\right\}$ ;
2. $\left\{{\Phi }_{n},\left(t\right)\right\}=\left\{\sqrt{2},sin,\left(\pi \left(n+\frac{1}{2}\right)t\right)\right\}$ , $n\in \left\{0,,,1,,,2,,,...\right\}$ ;
3. $\left\{{\Phi }_{n},\left(t\right)\right\}=\left\{1,,,\sqrt{2},cos,\left(\pi nt\right)\right\}$ , $n\in \left\{1,,,2,,,...\right\}$ ;
4. $\left\{{\Phi }_{n},\left(t\right)\right\}=\left\{\sqrt{2},sin,\left(\pi nt\right)\right\}$ , $n\in \left\{0,,,1,,,2,,,...\right\}$ .

Indeed, these orthonormal bases are obtained from the Fourier series on $\left(-2,2\right)$ (the first two) and on $\left(-1,1\right)$ (the last two) by appropriately imposing symmetries and hence are readily verified to becomplete and orthonormal on $\left(0,1\right)$ . If we choose a set of nonoverlapping rectangular window functions ${w}_{k}\left(t\right)$ such that ${\sum }_{k}{w}_{k}\left(t\right)=1$ for all $t\in \text{ℝ}$ , and define ${\chi }_{k,n}\left(t\right)={w}_{k}\left(t\right){\Phi }_{n}\left(t\right)$ , then, $\left\{{\chi }_{k,n},\left(t\right)\right\}$ is a local trigonometric basis for ${L}^{2}\left(\text{ℝ}\right)$ , for each of the four choices of $ph{i}_{n}\left(t\right)$ above.

## Construction of smooth windows

We know how to construct orthonormal trigonometric bases for disjoint temporal bins or intervals. Now we need to construct smooth windows ${w}_{k}\left(t\right)$ that when applied to cosines and sines retain orthonormality. An outline of the process is as follows:A unitary operation is applied that “unfolds” the discontinuities of all the local basis functions at the boundaries ofeach temporal bin. Unfolding leads to overlapping (unfolded) basis functions. However,since unfolding is unitary, the resulting functions still form an orthonormal basis. The unfolding operator is parameterizedby a function $r\left(t\right)$ that satisfies an algebraic constraint (which makes the operator unitary). The smoothness of the resulting basisfunctions depends on the smoothness of this underlying function $r\left(t\right)$ .

The function $r\left(t\right)$ , referred to as a rising cutoff function, satisfies the following conditions (see [link] ) :

${\left|r,\left(,t,\right)\right|}^{2}+{\left|r,\left(,-,t,\right)\right|}^{2}=1,\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\in \mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R};\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}r\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}t\le -1\hfill \\ 1,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}t\ge 1\hfill \end{array}\right)$

$r\left(t\right)$ is called a rising cutoff function because it rises from 0 to 1 in the interval $\left[-1,1\right]$ (note: it does not necessarily have to be monotone increasing). Multiplying a function by $r\left(t\right)$ would localize it to $\left[-1,\infty \right]$ . Every real-valued function $r\left(t\right)$ satisfying [link] is of the form $r\left(t\right)=sin\left(\theta \left(t\right)\right)$ where

$\theta \left(t\right)+\theta \left(-t\right)=\frac{\pi }{2}\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\in \mathrm{I}\phantom{\rule{-1.99997pt}{0ex}}\mathrm{R};\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}r\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}t\le -1\text{.}\hfill \\ \frac{\pi }{2},\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}t\ge 1\text{.}\hfill \end{array}\right)$

This ensures that $r\left(-t\right)=sin\left(\theta \left(-t\right)\right)=sin\left(\frac{\pi }{2}-\theta \left(t\right)\right)=cos\left(\theta \left(t\right)\right)$ and therefore ${r}^{2}\left(t\right)+{r}^{2}\left(-t\right)=1$ . One can easily construct arbitrarily smooth risingcutoff functions. We give one such recipe from [link] (p.105) . Start with a function

${r}_{\left[0\right]}\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}t\le -1\hfill \\ sin\left(\frac{\pi }{4},\left(1+t\right)\right),\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}-1

It is readily verified to be a rising cutoff function. Now recursively define ${r}_{\left[1\right]}\left(t\right),{r}_{\left[2\right]}\left(t\right),...$ as follows:

${r}_{\left[n+1\right]}\left(t\right)={r}_{\left[n\right]}\left(sin\left(\frac{\pi }{2}t\right)\right).$

Notice that ${r}_{\left[n\right]}\left(t\right)$ is a rising cutoff function for every $n$ . Moreover, by induction on $n$ it is easy to show that ${r}_{\left[n\right]}\left(t\right)\in {C}^{{2}^{n}-1}$ (it suffices to show that derivatives at $t=-1$ and $t=1$ exist and are zero up to order ${2}^{n}-1$ ).

## Folding and unfolding

Using a rising cutoff function $r\left(t\right)$ one can define the folding operator, $U$ , and its inverse, the unfolding operator ${U}^{☆}$ as follows:

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
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
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!