Universal Wavelet Generators for Dense Affine Sets

Definitions and mathematical statement

Use the positive affine group as a coordinate convention: \[ \mathcal A=(0,\infty)\times{\mathbb{R}},\qquad (a,b)(a',b')=(aa',b+ab'),\qquad (\pi(a,b)\psi)(x)=a^{-1/2}\psi((x-b)/a). \] For a countable affine parameter set $\Gamma\subset\mathcal A$, let $W(\psi,\Gamma)=(\pi(\gamma)\psi)_{\gamma\in\Gamma}$. It is a wavelet frame for $L^2({\mathbb{R}})$ if there are constants $0<A\leq B<\infty$ such that \[ A\|f\|_2^2\leq\sum_{\gamma\in\Gamma}|\langle f,\pi(\gamma)\psi\rangle|^2 \leq B\|f\|_2^2\qquad(f\in L^2({\mathbb{R}})). \] The source asks whether one generator works for every sufficiently dense affine set. Denoting the intended lower and upper affine densities by $D^-_{\mathrm{aff}}$ and $D^+_{\mathrm{aff}}$, its quantifier structure is \[ \exists\psi\in L^2({\mathbb{R}})\setminus\{0\}\quad \forall\Gamma\quad \left(D^-_{\mathrm{aff}}(\Gamma)>1,\ D^+_{\mathrm{aff}}(\Gamma)<\infty\right) \Longrightarrow W(\psi,\Gamma)\text{ is a frame for }L^2({\mathbb{R}}). \] Here $\Gamma$ ranges over the intended admissible affine parameter sets. The frame bounds may depend on $\Gamma$ and on the fixed $\psi$; uniform bounds across all such sets are not requested. The generator is chosen before $\Gamma$, so allowing a different generator for each set would weaken the question.

The supplied record does not define affine density, its normalization, test regions, translation convention, or the precise admissible class of sets. These are essential on the noncommutative affine group, and in particular to the numerical threshold one. The display therefore preserves the intended density predicate as an unspecified source definition; it does not substitute Euclidean density or an invented Haar-measure convention. The action above is an explicit standard coordinate convention to clarify wavelet frames and must be matched to the source's parameterization.

The supplied assessment checked 2026-08-21 reports that known necessary and sufficient density results impose additional regularity or geometric assumptions, and that the unrestricted universal statement remains unresolved in that check. Those additional hypotheses must not be silently imposed on the universal question.

Short English statement

Is there one wavelet generator that gives a frame for every affine parameter set whose lower affine density exceeds one and whose upper affine density is finite? The density convention and admissible sets must be recovered before this becomes a fully specified conjecture.

Sources

• Local UnsolvedMath registry, permanent ID 30000046, catalogue entry, Universal Wavelet Generators for Dense Affine Sets.
• Mini-Workshop: Wavelets and Frames (2004), Oberwolfach Reports, pp. 479--544, DOI https://doi.org/10.4171/owr/2004/10.
• Supplied reference "Weighted wavelet-frame density": https://doi.org/10.1007/BF02922055.
• Supplied reference "Wavelet-frame density and stability": https://doi.org/10.1016/S1063-5203(03)00049-6.
• Supplied reference "Affine density and wavelet frames": https://doi.org/10.1007/s00365-005-0620-y.
• Supplied literature triage checked 2026-08-21; no full author lists, theorem statements, or density definitions are reproduced.