Reconstructing Refinable Functions

Nathanael Berglnd In the summer of 2001 for a job I did an REU (Research Experience for Undergraduates) at Texas A&M University. The topic for the summer was Matrices and Wavelets. One of the fundamental building blocks of wavelets is a function called the scaling function. This function satisfies an equation known as a refinement equation. The set of all functions which satisfy a refinement equation is known as the set of refinable functions. These were the functions I focused my study on. Some of the questions asked included: Can you find a way to determine the class of all refinable functions? all continuous refinable functions? all refinable functions with compact support? etc. At the end of the summer I had found a classification theorem that answered these questions. I also wrote a software program to construct refinable functions based on the theorem.

	A refinable function is a function which is equal to the sum of scaled copies of itself, where the copies are contracted horizontally by 1/2, shifted left or right by multiples of 1/2 and scaled by real numbers. For example, the function on the left can be refined as: f(x) = f(2x) + f(2x - 1)
The function below is known as the "hat" function. Notice that it can be refined as: f(x) = 1/2 f(2x+1) + f(2x) + 1/2 f(2x-1)

	This function can be obtained from varying the coefficients between the two above functions, so that f(x) = (1-λ) f(2x + 1) + f(2x) + λ f(2x - 1), where 0 ≤ λ ≤ 1. It is possible to show that this function is continuous.
	The function on the left shows what happens if you allow λ to grow beyond 1. Note that this is no longer continuous. It is, however, still refinable over a set known as the dyadic numbers. Whether or not this function can be extended to all real numbers is a question I have not yet answered. Below is a function known as the Daubechies scaling function, so named because it was discovered by a mathematician named Ingrid Daubechies. It's refinement coefficients are:
Here are some more refinable functions I was able to discover. Can you guess what their refinement is?

For more information about the results I discovered, or to obtain software, see the links below. To better understand the software and what it does, I highly recommend you read the summary of results before running any of the programs.
summary.pdf - A summary of my results.
software.html - Describes the programs I have written.
paper.dvi - The full length paper detailing my results.

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