The Answer to Baggett's Problem is Affirmative

TL;DR

This paper proves that for a Parseval wavelet in L^2(R), the intersection of all its negative dilates' spaces is trivial, confirming a specific property related to wavelet dilations.

Contribution

It establishes that the intersection of negative dilates of a Parseval wavelet's span is zero, affirming Baggett's problem.

Findings

The intersection of negative dilates is the zero space.

The result confirms a conjecture about wavelet dilates.

Provides a rigorous proof for the property of Parseval wavelets.

Abstract

Let $\psi$ be a Parceval wavelet in $L^2 (\R)$ with the space of negative dilates $V(\psi)$. The intersection of the dilates $V(\psi)$ is the zero space. In other words, we have \begin{align*} \bigcap_{n\in\Z} D^n \overline{\textrm{span}}\{D^{\textrm{-}m} T^\ell \psi \mid m\geq 0, m,\ell\in\Z\}=\{0\}. \end{align*}