[Paper Review] Large classes of minimally supported frequency wavelets of L^2(\R) and H^2(\R)

Nicola Arcozzi, Biswaranjan Behera+1 more|ArXiv.org|Jul 17, 2002

Mathematical Analysis and Transform Methods4 references4 citations

TL;DR

This paper introduces a geometric construction method to generate large families of minimally supported frequency (MSF) wavelets for $L^2(\mathbb{R})$ and $H^2(\mathbb{R})$, using symmetric wavelet sets derived from dyadic polygons. It proves the existence of uncountably many symmetric wavelet sets, including those with non-dyadic endpoints and wavelets whose Fourier transforms do not vanish near the origin, resolving open questions by Hernández and Garrigós.

ABSTRACT

We introduce a method to construct large classes of MSF wavelets of the Hardy space H^2(\R) and symmetric MSF wavelets of L^2(\R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L^2(\R) and H^2(\R). We also enumerate all symmetric wavelets of L^2(\R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H^2(\R). Finally, we construct families of MSF wavelets of L^2(\R) whose Fourier transform does not vanish in any neighbourhood of the origin.

Motivation & Objective

• To develop a systematic method for constructing large classes of MSF wavelets in $L^2(\mathbb{R})$ and $H^2(\mathbb{R})$.
• To classify symmetric wavelet sets with finitely many intervals in the positive real line for all $n \geq 1$.
• To resolve open questions regarding the existence of wavelets with non-vanishing Fourier transforms near the origin and wavelets with non-dyadic endpoints.
• To demonstrate the uncountability of symmetric wavelet sets and interval wavelet sets in both $L^2(\mathbb{R})$ and $H^2(\mathbb{R})$.

Proposed method

• Construct symmetric wavelet sets by associating them with polygons whose vertices lie on a dyadic lattice, enabling parameterized families for any $n \geq 1$.
• Use geometric conditions (T) and (D) from Theorem 1: $\coprod_{n\in\mathbb{Z}}(K+n) = \mathbb{R}$ a.e. and $\coprod_{n\in\mathbb{Z}}2^nK = \mathbb{R}$ a.e. to ensure wavelet set validity.
• Define wavelet sets $K_{n,\epsilon}$ via recursive interval constructions involving $E_l$, $F_l$, $G_l$, $H_l$, and fixed intervals $R_2$, $S_1$, $S_2$, $S_3$, $L_2$, $T_1$, $T_2$, $T_3$, with a real parameter $\epsilon$.
• Prove dilation and translation equivalence between $K_{n,\epsilon}$ and a base wavelet set $K_n$, ensuring $K_{n,\epsilon}$ satisfies the wavelet set conditions.
• Use symmetry and recursive inclusion of intervals to ensure $K_{n,\epsilon}$ is bounded and has 0 as an accumulation point.
• Leverage the parameter $\epsilon \in (0, e_n/4)$ to generate uncountably many distinct symmetric wavelet sets, including those with non-dyadic endpoints.

Experimental results

Research questions

• RQ1 Are there uncountably many symmetric wavelet sets in $L^2(\mathbb{R})$ for $n=3$ intervals in $K^+$?
• RQ2 Can MSF wavelets be constructed such that their Fourier transform does not vanish in any neighborhood of the origin?
• RQ3 Do symmetric wavelet sets exist with non-dyadic endpoints, i.e., not of the form $p/2^q$?
• RQ4 Is there a complete classification of symmetric wavelet sets for $n \geq 3$?
• RQ5 Can the construction method yield wavelets in the Hardy space $H^2(\mathbb{R})$ with specific interval structures?

Key findings

• For $n=3$, the family of symmetric wavelet sets defined by $K_a^+ = [a, 1/2] \cup [1-a, 2a] \cup [1, 2(1-a)]$ with $a \in (1/3, 1/2)$ provides uncountably many such sets.
• The wavelet sets $K_{n,\epsilon}$ constructed for $n \geq 2$ and $0 < \epsilon < e_n/4$ are bounded and have 0 as an accumulation point.
• The Fourier transform of the wavelet $\psi_{n,\epsilon}$ associated with $K_{n,\epsilon}$ is even and does not vanish in any neighborhood of the origin.
• The paper constructs a family of symmetric wavelet sets where endpoints depend continuously on a real parameter $a$, proving uncountability beyond $n=2$.
• For $n=1$, the only symmetric wavelet set is the Shannon set $K^+ = [1/2, 1]$.
• The paper enumerates all symmetric wavelets of $L^2(\mathbb{R})$ with at most three intervals in the positive axis and 3-interval wavelet sets of $H^2(\mathbb{R})$.