Variable Bandwidth for Locally Oscillatory Signals

Definitions and mathematical statement

Let $L^2({\mathbb{R}})$ denote the space of Lebesgue-measurable scalar-valued functions satisfying \[ \int_{{\mathbb{R}}}|f(t)|^2\,{\,\mathrm{d}} t<\infty, \] with functions identified when they agree almost everywhere. The source does not specify whether the scalar field is ${\mathbb{R}}$ or ${\mathbb{C}}$. A signal in this problem is an element $f\in L^2({\mathbb{R}})$.

Research question. Develop a rigorous notion of variable bandwidth for signals $f\in L^2({\mathbb{R}})$ that captures local frequency variation and supports appropriate sampling, density, and analyticity properties.

Unresolved formulation. The supplied record states a request for a mathematical framework, rather than a conjecture with specified hypotheses and conclusion. It does not define local frequency variation, prescribe the mathematical object representing variable bandwidth, or specify the sampling, density, or analyticity properties required. Nor does it state whether the framework must apply to every element of $L^2({\mathbb{R}})$ or to specified subclasses. Since an $L^2$ equivalence class has no intrinsic point values, any formulation using point samples must specify representatives or additional regularity. The relevant sampling sets, meaning of density, and domain and meaning of analyticity also remain unspecified. Consequently, the supplied evidence does not determine a unique quantified assertion or a criterion by which a proposed framework would resolve the question.

Short English statement

The question asks for a rigorous framework describing how a square-integrable signal's frequency content varies locally, together with suitable sampling, density, and analyticity properties. The source leaves these requirements unspecified, so this is a research-program statement requiring further mathematical choices. Its supplied status is partially solved; that status is not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $30002944$, title Variable Bandwidth for Locally Oscillatory Signals, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report attribution: Applied Harmonic Analysis and Sparse Approximation (2016), Oberwolfach Reports, pp. 2189--2263, DOI 10.4171/owr/2015/38. https://doi.org/10.4171/owr/2015/38. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied citation labels the report 2016, whereas the DOI contains 2015 and the registry records proposed year 2015; these details are retained without resolving their relationship.
• Supplied literature reference: Groechenig and Klotz. https://arxiv.org/abs/1512.06663.
• Supplied literature reference: Aceska and Feichtinger 2011. https://www.sciencedirect.com/science/article/pii/S0022247X11003763.
• Supplied literature reference: Andreolli and Groechenig 2023. https://arxiv.org/abs/2305.17290.
• Supplied literature assessment, checked at 2026-08-21: the record labels the problem partially solved and reports that rigorous spectral and time-frequency definitions of variable bandwidth have sampling theories, while multiple frameworks exist rather than one canonical notion. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are supplied editorial assertions; this draft does not independently verify the report or linked references.