-refinable

.

where denotes convolution.  Below are some plots of some of the first few B-Splines.  The first two should look familiar as they are the simplest examples of refinable functions that I used in my first post.

As an example I will go through the computation of using the definition given above.

At this point since is zero whenever then one only need integrate over the interval so that,

.

Now for a quick change of variables letting so that and

.

Now we break into pieces

.

Cleaning everything up, we have

.

The other B-Splines can be generated similarly.

]]> http://jmeuser.wordpress.com/2009/09/10/uniform-b-splines-are-refinable/feed/ 2 jmeuser phi Hat B3 http://jmeuser.wordpress.com/2009/09/03/fields-and-field-properties/ http://jmeuser.wordpress.com/2009/09/03/fields-and-field-properties/#comments Thu, 03 Sep 2009 02:00:08 +0000 jmeuser http://jmeuser.wordpress.com/?p=191 ]]> Any discussion of Real Analysis requires a short introduction to fields and their basic properties.  Simply put, a field is a set like which follows the normal rules of addition and multiplication taught in elementary school; things like distributivity, commutativity, and associativity.  With a well defined concept of field we can prove general properties that all fields must have, and then rejoice at the additional structure added to any set on which we can define operations of multiplication and addition which satisfy the field axioms.  So without further ado:

Definition: (Field) A field is a set with two operations defined on it called multiplication, denoted by ““, and addition, denoted by ““, which satisfy the following axioms:

Addition

1. (Closure) If then
2. (Commutativity)   for all
3. (Associativity) for all
4. (Identity) There exists an element such that for all
5. (Inverse) For every there exists such that

Multiplication

1. (Closure) If then
2. (Commutativity) for all
3. (Associativity)   for all
4. (Identity) There exists with such that for all
5. (Inverse) For every with there exists such that .
6. (Distributivity) for all .

So that’s all fine and dandy, especially since you already know all of these axioms from the real numbers, but now that we are armed with a formal definition of a field we can start to prove formally why the identity element, and inverse element must be unique.

Proposition:

1. If then
2. If then
3. If then
4. 

Proof: (1)

.

(2)

(3)

(4)

.

From (1) we get the good old cancelation law which we’ve used since middle school algebra to whittle down our equations and solve for the infamous unknown “x”.

(2) says that the identity element is the same for all and therefore unique.  Another way to prove that the identity element is unique is to assume that it isn’t.  That is assume that there are two distinct identity elements and , then by the identity field axiom .  This contradicts our assumption that there were two distinct identity elements, so there must only be one.

With (3) we learn that the inverse elements are unique, that for each element there is a unique which is the inverse of .  (4) Says that the inverse of the inverse of is just the same thing as .

]]> http://jmeuser.wordpress.com/2009/09/03/fields-and-field-properties/feed/ 0 jmeuser http://jmeuser.wordpress.com/2009/08/31/a-math-joke/ http://jmeuser.wordpress.com/2009/08/31/a-math-joke/#comments Mon, 31 Aug 2009 20:54:08 +0000 jmeuser http://jmeuser.wordpress.com/?p=194 ]]> A little joke that popped into my head during a math class.

Definition: (Ordered Set) A set is an ordered set if there exists a relation “” on such that

(1) only one of the following are true: , ,

(2) if and then .

Definition: (Bounded above) Let be an ordered set and then if there exists such that latex  for all then is bounded above and is an upper bound of .

Lower bounds are defined by swapping with .

Definition: (Least Upper Bound/Supremum ) Let be an ordered set and then is the least upper bound or supremum of if

(1) is an upper bound of

(2) for any upper bound of , .

The supremum is denoted by .  The greatest lower bound or infimum of a set is defined in the obvious manner and denoted by .

Definition: (Least-Upper-Bound Property) An ordered set is said to have the least-upper-bound property if  the supremum of any non-empty subset of with an upper bound exists in .   The analogous greatest-lower-bound property is defined in the obvious way.

The following thoeorem shows that if a set has the least-upper-bound proerty then it also has the great-lower-bound property.

Theorem: Suppose is an ordered set with the least-upper-bound property, , is not empty, and is bounded below.  Let be the set of all lower bounds of then .

Proof: Since is bounded below it has at least one lower bound so is not empty. Every element in is an upper bound of thus by the least-upper-bound property there exists .
If then since is not an upper bound of .  Thus for all so that is a lower bound of B and .  If then since is an upper bound.  Thus is a lower bound of , , and any is not a lower bound of so .  Therefore .

Proposition: Let be a refinement sequence and define   then

1) , .

2) and , .

Proof: (1) Let then and thus

therefore .

(2) Let then

therefore .

So if you fix a refinement sequence and collect all the functions in a vector space, like , then forms a subspace (the zero function is refinable for any refinement sequence).

Proposition: If is differentiable and refinable with refinement sequence then is refinable with refinement sequence .

Proof: Simply take the derivative over the finite sum and apply the chain rule,

.

Therefore is refinable with refinement sequence .

There is a similar property for integration that you might want to work out as a little puzzle.  In this case assume that for some there exists a function and prove that is refinable with a refinement sequences whose coefficients are half of the refinement sequence of .

Finally we look at one of the more interesting properties of refinable functions which provides a way of constructing a large number of refinable functions from known refinable functions.

Definition: (Convolution)  The convolution of two functions and is defined as follows,

.

Proposition: If and are refinable and integrable over then is refinable.

Proof: Assume ‘s refinement sequence is and ‘s refinement sequence is then,

Let then and substituting,

Let , then

That concludes this post.  Next time I’ll give some specific examples of a set of functions created using this method of convolution and maybe present an alternate definition of Refinable Functions using the Fourier transformation.  Feel free to leave questions and comments in the comments section; I’ll answer any questions as quickly as possible.

]]> http://jmeuser.wordpress.com/2009/07/23/some-properties-of-refinable-functions/feed/ 0 jmeuser http://jmeuser.wordpress.com/2009/07/18/introduction-to-refinable-functions/ http://jmeuser.wordpress.com/2009/07/18/introduction-to-refinable-functions/#comments Sat, 18 Jul 2009 10:07:05 +0000 jmeuser http://jmeuser.wordpress.com/?p=8 ]]> During the summers of 2008 and 2009 I participated in a NSF funded REU at Texas A&M.  Under the guidance of Dr. David Larson I researched, among other things, refinable functions.  Here and in following posts I will present some of the results obtained during these mathematically stimulating summers.

Definition: A function is finitely m-refinable if there exist integers with   and with and such that for all .

The set is called the scaling (or refinement) sequence of and is called the scaling equation.

For the sake of simplicity we will consider only finitely 2-refinable functions.  So when we say a function is refinable, unless otherwise noted, we mean a finitely 2-refinable function.

Examples:

Our first example is the characteristic function of also known as the scaling function of the Haar Wavelet defined so that

    and graphed below.

It has the refinement sequence so that .  The first term is graphed in red the and the second term graphed in green below.

The second example is the hat function defined so that

and so called because of it’s shape shown below.

It has the refinement sequence and so that .  Each term in the refinement sequence is graphed below.

The final example of a refinable function is actually a whole class of functions that are refinable: polynomials.  While this might not be obvious at first (it wasn’t for me) take a few minutes and work it out on your own.  It’s a fun little puzzle and provides you with a better understanding of refinable functions and their interesting properties.  In my next post I will prove some basic properties of refinable functions.

Mathematica: The images used in this post were created using Mathematica and the following code:

Clear[\[Phi]]
\[Phi][x_] := \[Piecewise] {
 {1, 0 <= x < 1},
 {0, True}
 }
Plot[\[Phi][x], {x, -.5, 1.5}, PlotStyle -> Thick,
 PlotRange -> {0, 1.5}]
Plot[{\[Phi][2 x], \[Phi][2 x - 1]}, {x, -.5, 1.5},
 PlotStyle -> {{Thick, Dashed, Red}, {Thick, Dashed, Green}},
 PlotRange -> {0, 1.5}]

Clear[Hat]
Hat[x_] := \[Piecewise] {
 {x, 0 <= x < 1},
 {2 - x, 1 <= x < 2},
 {0, True}
 }
Hat[x_] := If[0 <= x < 1, x, If[1 <= x < 2, 2 - x, 0]]
Plot[Hat[x], {x, -.5, 2.5}, PlotStyle -> Thick,
 PlotRange -> {0, 1.2}]
Plot[{1/2 Hat[2 x], Hat[2 x - 1], 1/2 Hat[2 x - 2]}, {x, -.5, 2.5},
 PlotStyle -> Thick, PlotRange -> {0, 1.2}]
Plot[.5 Hat[2 x] + Hat[2 x - 1] + .5 Hat[2 x - 2], {x, -.5, 2.5},
 PlotStyle -> Thick]