Path connectivity of orthonormal wavelets

Public text index for scholarly discovery. It mirrors theorem, lemma, proposition and related statements from this immutable version and does not imply human verification.

1. Theorem 1.1
   PF:2026.000010/v1/THM-1.1

   For any $\psi_0,\psi_1\in\mathcal{W}$, there is an $\textup{L}^2$-continuous map $\gamma\colon[0,1]\to\mathcal{W}$ with $\gamma(0)=\psi_0$ and $\gamma(1)=\psi_1$.

2. Lemma 2.1
   PF:2026.000010/v1/LEM-2.1

   For $\psi\in\mathcal W$, put \[ W_j:=\overline{\operatorname{span}}\{D^jT_k\psi:k\in\Z\}, \qquad V:=\bigoplus_{j<0}W_j. \] The \emph{core} $V$ is shift invariant and satisfies V\subset DV,\qquad \overline{\bigcup_{n\geq0}D^nV}=\textup{L}^2(\R),\qquad \bigcap_{n\geq0}D^{-n}V=\{0\}. Its fiber dimension is the wavelet dimension function m(\xi):=m_\psi(\xi):=\dim J_V(\xi) =\sum_{j\geq1}\sum_{k\in\Z} |\widehat\psi(2^j(\xi+k))|^2, \qquad \int_Km(\xi)\,d\xi=1. Thus $m$ is finite almost everywhere. Moreover, with $J:=J_V$, $F:=J_{DV}$, and $u:=\mathcal T\psi$, F(\xi)=J(\xi)\oplus\C u(\xi),\qquad \|u(\xi)\|_H=1.

3. Lemma 2.2
   PF:2026.000010/v1/LEM-2.2

   Suppose $V$ is shift invariant, has finite-dimensional fibers almost everywhere, and satisfies \[ V\subset DV,\qquad \overline{\bigcup_{n\geq0}D^nV}=\textup{L}^2(\R). \] If $W:=DV\ominus V$ has one-dimensional fibers, then every measurable unit vector field $u(\xi)\in J_W(\xi)$ determines an orthonormal wavelet $\psi:=\mathcal T^{-1}u$ with core $V$.

4. Lemma 3.1
   PF:2026.000010/v1/LEM-3.1

   Let $1\geq a_1>a_2>\cdots>0$ and $A_t:=\operatorname{diag}(a_j^t)$ on $\ell^2(\N)$, $t\geq0$. For each finite-dimensional subspace $J$, the projections $P_{A_tJ}$ are norm continuous and converge in operator norm as $t\to\infty$ to a coordinate projection of rank $\dim J$. Its range $J_\infty$ contains $e_1$ whenever the first coordinate functional is nonzero on $J$. If $F=J\oplus\C u$ with $u\perp J$ and $\|u\|=1$, then w_t:=\frac{(I-P_{A_tJ})A_tu}{\|(I-P_{A_tJ})A_tu\|} is well defined and continuous, starts at $u$, and converges to a unimodular multiple of the unique coordinate vector in $F_\infty\ominus J_\infty$.

5. Lemma 3.2
   PF:2026.000010/v1/LEM-3.2

   For every closed shift-invariant space $U$ and every finite $t\geq0$, \overline{L_tDU}=D\,\overline{L_tU}.

6. Lemma 4.1
   PF:2026.000010/v1/LEM-4.1

   For every $\psi\in\mathcal W$, there are a wavelet set $E$ and an $\textup{L}^2$-continuous map $\alpha:[0,1]\to\mathcal W$ such that \[ \alpha(0)=\psi,\qquad \alpha(1)=\mathcal F^{-1}\mathbbm1_E,\qquad m_{\alpha(s)}=m_\psi\quad(0\leq s\leq1). \]