Lower Density of Wave-Packet Frames

Definitions and mathematical statement

For $\psi\in L^2({\mathbb{R}})$, let $\mathcal W(\psi,M)=\{\psi_m:m\in M\}$ be the wave-packet system indexed by the discrete parameter set $M$ in the source's dilation--translation--modulation parameter space. The standard unitary operations are \[ T_bf(x)=f(x-b),\qquad E_cf(x)=e^{2\pi\sqrt{-1}cx}f(x),\qquad D_af(x)=a^{-1/2}f(x/a)\quad(a>0). \] The ordering of these operations and the corresponding parameter-space group law must agree with the referenced construction; they are not specified in the extract.

Having an upper frame bound means the Bessel condition \[ \exists B<\infty\ \forall f\in L^2({\mathbb{R}}),\qquad \sum_{m\in M}|\langle f,\psi_m\rangle|^2\le B\|f\|_2^2. \] The source conjecture is that this condition forces \[ \dim^-(M)=0, \] where $\dim^-$ is the lower dimension used for wave-packet parameter sets in the cited work. The extract does not define its averaging regions, parameter-space geometry, or limiting convention. Those data are essential and cannot be replaced by the Hausdorff dimension of a discrete subset or an arbitrary Euclidean density convention.

The verified sentence requires only an upper bound; no positive lower frame bound, completeness or full-frame assumption is added. It also does not explicitly exclude $\psi=0$. A nonzero-generator convention is essential to check: the zero generator makes the upper bound automatic for every parameter set and cannot impose a nontrivial universal dimension restriction. This is an unresolved literal qualification, not a silently repaired hypothesis.

The literature assessment checked on 21 August 2026 reports that Czaja--Kutyniok--Speegle prove a dichotomy and handle special parameter sets, but found no later proof of the asserted zero lower dimension. The dichotomy's precise alternatives are not supplied, and its use of "frame" in the assessment does not justify strengthening the displayed Bessel-only hypothesis. The open label is dated literature triage requiring independent verification.

Short English statement

Does an upper frame bound force zero lower parameter-set dimension for a wave-packet system? The target uses the source's wave-packet geometry and needs its precise dimension definition and nonzero-generator convention; an upper bound alone is the stated hypothesis.

Sources

• Mini-Workshop: Wavelets and Frames (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/10, pp. 479--544, https://doi.org/10.4171/owr/2004/10.
• Literature assessment checked 21 August 2026: Czaja--Kutyniok--Speegle, https://doi.org/10.1016/j.acha.2005.04.002; "Oberwolfach report," https://ems.press/content/serial-article-files/45879.
• Local UnsolvedMath registry, permanent ID $30000039$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Its clean statement is labelled corrected and verified by restoring the generator, function space, parameter set and numerical value.