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We will now look more closely at the basic scaling function and wavelet to see when they exist and what their properties are [link] , [link] , [link] , [link] , [link] , [link] , [link] . Using the same approach that is used in the theory of differentialequations, we will examine the properties of φ ( t ) by considering the equation of which it is a solution. The basic recursion [link] that comes from the multiresolution formulation is

φ ( t ) = n h ( n ) 2 φ ( 2 t - n )

with h ( n ) being the scaling coefficients and φ ( t ) being the scaling function which satisfies this equation which is sometimes calledthe refinement equation , the dilation equation , or the multiresolution analysis equation (MRA).

In order to state the properties accurately, some care has to be taken in specifying just what classes of functions are being considered or areallowed. We will attempt to walk a fine line to present enough detail to be correct but not so much as to obscure the main ideas and results. A fewof these ideas were presented in Section: Signal Spaces and a few more will be given in the next section. A more complete discussion can be foundin [link] , in the introductions to [link] , [link] , [link] , or in any book on function analysis.

Tools and definitions

Signal classes

There are three classes of signals that we will be using. The most basic is called L 2 ( R ) which contains all functions which have a finite, well-defined integral of the square: f L 2 | f ( t ) | 2 d t = E < . This class is important because it is a generalization of normal Euclideangeometry and because it gives a simple representation of the energy in a signal.

The next most basic class is L 1 ( R ) , which requires a finite integral of the absolute value of the function: f L 1 | f ( t ) | d t = K < . This class is important because one may interchange infinite summations and integrations with these functions although not necessarily with L 2 functions. These classes of function spaces can be generalized to those with | f ( t ) | p d t = K < and designated L p .

A more general class of signals than any L p space contains what are called distributions . These are generalized functions which are not defined by their having “values" but by the value of an “inner product"with a normal function. An example of a distribution would be the Dirac delta function δ ( t ) where it is defined by the property: f ( T ) = f ( t ) δ ( t - T ) d t .

Another detail to keep in mind is that the integrals used in these definitions are Lebesque integrals which are somewhat more general than the basic Riemann integral. The value of a Lebesque integral is notaffected by values of the function over any countable set of values of its argument (or, more generally, a set of measure zero). A function definedas one on the rationals and zero on the irrationals would have a zero Lebesque integral. As a result of this, properties derived using measuretheory and Lebesque integrals are sometime said to be true “almost everywhere," meaning they may not be true over a set of measure zero.

Many of these ideas of function spaces, distributions, Lebesque measure, etc. came out of the early study of Fourier series and transforms. It isinteresting that they are also important in the theory of wavelets. As with Fourier theory, one can often ignore the signal space classes and canuse distributions as if they were functions, but there are some cases where these ideas are crucial. For an introductory reading of this bookor of the literature, one can usually skip over the signal space designation or assume Riemann integrals. However, when a contradiction orparadox seems to arise, its resolution will probably require these details.

Questions & Answers

An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
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
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
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salma
Commplementary angles
Idrissa Reply
hello
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Tamia
hii
Uday
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salma
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Ayuba
Hello
opoku
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Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
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
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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
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
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
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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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