Support Uncertainty for Finite Cyclic Short-Time Fourier Transforms

Definitions and mathematical statement

Let $G$ be a finite cyclic group, written additively, and put $n=|G|$. Let ${\mathbb{C}}^G$ be the space of complex-valued functions on $G$, and let $\widehat G$ be its character group. For a function $f$ on a finite set $S$, define \[ \operatorname{supp}(f)=\{s\in S:f(s)\ne0\},\qquad \|f\|_0=|\operatorname{supp}(f)|. \] Using standard unnormalized conventions, define the Fourier transform and the short-time Fourier transform with window $\varphi\in{\mathbb{C}}^G$ by \[ \widehat x(\chi)=\sum_{t\in G}x(t)\overline{\chi(t)},\qquad \chi\in\widehat G, \] \[ V_\varphi x(a,\chi)=\sum_{t\in G}x(t)\overline{\varphi(t-a)}\,\overline{\chi(t)},\qquad (a,\chi)\in G\times\widehat G. \] Thus $\|V_\varphi x\|_0$ counts nonzero coefficients among the $n^2$ time-frequency positions. These conventions supply definitions omitted from the record; nonzero normalization factors do not change support sizes.

The supplied question asks whether, for every finite cyclic group $G$, there exists $\varphi\in{\mathbb{C}}^G\setminus\{0\}$ such that \[ \left\{\left(\|x\|_0,\|V_\varphi x\|_0\right):x\in{\mathbb{C}}^G\right\} = \left\{\left(\|y\|_0,n^2-n+\|\widehat y\|_0\right):y\in{\mathbb{C}}^G\right\}. \] This is equality of attainable sets of pairs; it does not require the same signal to realize corresponding pairs on both sides.

Unresolved boundary qualification. The supplied quantifiers include the zero signal. The unique pair with first coordinate zero on the left is $(0,0)$, whereas the unique such pair on the right is $(0,n^2-n)$. These differ when $n>1$. Consequently, the literal formulation has a zero-signal incompatibility with its presentation as an open uncertainty question. Whether the intended statement excludes zero signals on both sides, treats the zero pair separately, or uses another qualification requires source review. No such repair is adopted here.

Short English statement

For each finite cyclic group, the question seeks a nonzero window whose short-time Fourier transform realizes exactly the support-size pairs obtained by shifting Fourier support sizes by $|G|^2-|G|$. The supplied formulation includes zero signals, creating an unresolved boundary inconsistency for $|G|>1$. Its supplied open status and publication-verification label are not independent mathematical verification.

Sources

• Local UnsolvedMath problem registry, permanent ID $30002569$, source code catalogue entry, title Support Uncertainty for Finite Cyclic Short-Time Fourier Transforms. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the proposed year is recorded as 2014.
• Supplied report citation: Mini-Workshop: Mathematical Physics meets Sparse Recovery (2015), Oberwolfach Reports, pp. 1047--1073, DOI 10.4171/owr/2014/18. https://doi.org/10.4171/owr/2014/18. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2015 designation and the record's 2014 report/proposal designations are retained pending verification.
• Supplied literature source: Oberwolfach Report 2014/18. https://doi.org/10.4171/owr/2014/18.
• Supplied literature source: Krahmer--Pfander--Rashkov, Dagstuhl Proceedings. https://doi.org/10.4230/DagSemProc.08492.4. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that generic general-linear-position and prime-order results do not establish the full support-pair equality for every cyclic group. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is a supplied editorial qualification; the cited sources have not been independently consulted for this draft.