The Answer to Baggett's Problem is Affirmative

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*}