Open Problems in Wavelet Theory — Conjecture 2

Definitions and mathematical statement

Let $H=L^2({\mathbb{R}})$, equipped with the norm $\|f\|_2=(\int_{{\mathbb{R}}}|f(x)|^2\,{\,\mathrm{d}} x)^{1/2}$. Write $\mathcal W\subseteq H$ for the class called "o.n. wavelets" in the supplied source, and write $\mathscr D_\psi$ for the wavelet dimension function assigned to $\psi\in\mathcal W$. The supplied record does not define this class or assignment, specify the scalar field, or reproduce conditions $(\mathrm{D1})$--$(\mathrm{D4})$. These source-dependent ingredients remain unresolved.

For each wavelet dimension function $\mathscr D$ satisfying the source's conditions $(\mathrm{D1})$--$(\mathrm{D4})$, put \[ \mathcal W_{\mathscr D}=\{\psi\in\mathcal W:\mathscr D_\psi=\mathscr D\}. \] The supplied Conjecture 2 asserts that $\mathcal W_{\mathscr D}$ is path connected in the $L^2$-norm topology. Explicitly, for every such $\mathscr D$ and every $\psi_0,\psi_1\in\mathcal W_{\mathscr D}$, there exists a map $\gamma:[0,1]\to H$, continuous in the $L^2$ norm, such that \[ \gamma(0)=\psi_0,\qquad \gamma(1)=\psi_1,\qquad \gamma(t)\in\mathcal W,\quad \mathscr D_{\gamma(t)}=\mathscr D \quad\text{for every }t\in[0,1]. \] Here equality of dimension functions retains the source's unspecified convention; the record does not establish whether it means pointwise equality or equality almost everywhere. The omitted definitions and conditions must be recovered before this becomes a fully specified mathematical statement. No additional nonemptiness assertion is inferred from the omitted conditions.

Short English statement

For any fixed admissible wavelet dimension function, the conjecture says that any two orthonormal wavelets having that dimension function can be joined by an $L^2({\mathbb{R}})$-continuous path of orthonormal wavelets with the same dimension function throughout. The admissibility conditions and dimension-function definition are missing from the supplied record.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $801163$, exact source problem code catalogue entry, title "Open Problems in Wavelet Theory --- Conjecture 2." The registry attributes the problem to Bownik and Rzeszotnik, with proposed year 2019, and labels it open. This is a supplied status label, not an independently verified mathematical or literature assessment.
• Bownik and Rzeszotnik, Open Problems in Wavelet Theory (2019), Conjecture 2. Supplied source URL: https://arxiv.org/abs/1905.08230. The supplied provenance records access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance statements, not claims of independent source access here.