Summary of Projects - REU 2001

Sequences of Refinable Functions by Sodja Cole

Definition: A refinable function is one where there is a functional relationship between the function and its half-integer translates, i.e.

f(x) = sum_k a^k f(x-2k)

Here, the sequence, .. a_-1, a₀, a₁, a₂, ... could be infinite in length. The sequence a_k is called a refining sequence for f. Refinablility is one of the key defining properties of wavelets and is heavily used in the coding schemes for the decomposition and reconstruction algorithms in signal analysis.

Here, we investigate the preservation of refinability under various types of limiting operations. The following theorems are proved.

Theorem 1: Suppose that the sequence a_k belongs to "little L¹", i.e. that sum_k |a_k| is finite. Suppose f_n is a sequence of refinable functions, each with the refining sequence a_k. If f_n converges uniformly to f, then f is refinable with refining sequence a_k.

Theorem 2: Suppose that f_n is refinable with refining sequence Aⁿ= aⁿ_k , k = 1, 2, ... is a refining sequence for each n=1, 2, ... Suppose that there is a sequence B = b₁, b₂, ... of nonnegative numbers and which is in little L¹ (i.e. sum_k b_k is finite) with |f_n (x)|| aⁿ_k| less than or equal to b_k for all x in R. If f_n converges pointwise to f and aⁿ_k converges to alpha_k as n -> infinity (for each k), then f is refinable with refining sequence alpha_k.

Theorem 3: Let p be at least 1 (but finite) and suppose f_n is a sequence of refinable functions in L^p which converge weakly to f. Suppose Aⁿ=aⁿ_k, k = 1, 2, ... is the refining sequence for f_n and that Aⁿ converges to alpha in little L¹ as n -> infinity. Then f is refinable with refining sequence alpha_k.

On Interpolation Pairs of Shannon Wavelet Sets by Quoc Le Gia

A measurable set E is called a wavelet set if the characteristic function on E is the Fourier transform of a wavelet function. Two measurable sets, E and F, are 2Pi congruent if there is a measurable bijection f:E->F such that f(s)=s mod 2Pi for each s in E. One of the key results of Larson and Dai is that a set E is a wavelet set if and only if the following two conditions hold:

• E is 2Pi congruent to the interval [0,2Pi)
• 2^j E, as j ranges over Z is a partition of R.

In such a case, E and F is called an interpolation pair. If E and F is an interpolation pair, then the interpolation map f between them can be extended to the whole real line by defining f(x) = 2ⁿ f(2^-n x) where n is the unique integer where x belongs to 2ⁿ E.

Let a be a number between -Pi and Pi. Define

E_a = [-2Pi+2a, -Pi +a) union [Pi+a, 2Pi+2a)

This set is called a Shannon wavelet set. Here, we investigate necessary and sufficient condtions for two generalized Shannon wavelet sets to be an interpolation pair.

The main result is the following theorem.

Theorem: Let Omega be the parallelogram with vertices (-Pi,Pi), (Pi/3,-Pi/3), (Pi,Pi) and (-Pi/3, Pi/3). Then E_a and E_b is an interpolation pair if and only if (a,b) belongs to Omega.

The following theorem is also shown.

Theorem: Let E₁, E₂, ... E_n be wavelet sets so that each consecutive pair is an interpolation pair with interpolation map f_k: E_k -> E_k+1, k=1..n-1. Let f_n be the interpolation map between E_n and E₁. Then the following two conditions are equivalent:

• Any pair of sets from E_k, k=1..n form a pairwise interpolation pair.
• The interpolation maps pairwise commute.

An Example of a Continuous Finitely Refinable Function with Compact Support. by Nathanael Berglund

Here, we investigate when a refinable function with a finite refining sequence is continuous. We specialize to the case of a refinable function with refining sequence

a_-1 = lambda, a₀ =1, a₁ = 1-lambda

where 0 < lambda < 1.

The main theorem is the following:

Theorem: A refinable function f with refining sequence given above is continuous if f(-1)=0, f(0)=1, f(1)=0.

Other theorems on refinable functions are also given. For example it is shown that the zero is the only function with refining sequence given by all zeros. Also suppose the refining sequence starts at a and ends at b (so a_a is the first nonzero coefficient and a_b is the last). Then such refinable functions are determined their values on

(a-2,a-1] union {a} union [a+1, a+2) if a=b [a,b) if a is less than b

Additional theorems are given which state that such refinable functions can be classified according to their values on these intervals.

The Theory of Finite Interval Multiplicity Functions by Darren Rhea and Scott Armstrong

First, a definition of a multiplicity function:

Definition: A multiplicity function m:R -> N is a measurable function such that

• m(x)=m(x+1)
• m(x)+1=m(x/2)+m(x/2+1/2)
• Integral of m over [0,1] is 1
• liminf m(x/2^n) >=1 as n -> infinity, almost everywhere in (x).

Definition: A wavelet set W has the property that 2ⁿ W, n in Z and W+n, n in Z are partitions of R.

Definition: A generalized scaling set E is a measurable set such that

• E is contained in 2E
• Lim chi_E (2^-n x) =1 a.e. where chi is the characteristic function
• the measure of E is 1.
• m_E (x) = sum_k chi_E (x+k) is a multiplicity function.

We show the following theorem.

Theorem: Any multiplicity function which is a step function with a finite number of discontinuities in [0,1] must have its discontinuities at rationals with an odd denominator.