Critical Sampling Density Beyond Beurling Density

Definitions and mathematical statement

Source question. For settings such as Bergman spaces and wavelet frames where Beurling density is not the correct notion, does a critical density separating sampling from interpolation exist?

Here sampling refers to stable recovery from sampling data, and interpolation refers to realization of prescribed admissible data. Their precise meanings depend on the function space or frame system, the sampling or coefficient functionals, and the norm on the data space. These ingredients are not specified in the supplied record.

The requested object is a suitable density notion and an associated critical threshold distinguishing sampling configurations from interpolation configurations in the intended setting. The source does not define that density, the configurations to which it applies, or the mathematical meaning of "separating." In particular, it does not say whether the threshold supplies necessary conditions, sufficient conditions, or characterizations, nor what happens at equality. Thus no particular system of density inequalities is asserted here.

Unresolved scope. No particular Bergman space, underlying domain, weight, wavelet family, parameter space, or class of configurations is specified. The examples do not determine whether the intended assertion concerns every such setting or selected settings. The qualification that Beurling density is not the correct notion is retained as part of the source question, without extending it to all Bergman spaces or all wavelet frames. Additional hypotheses and a definition of the intended density criterion are required for a fully quantified mathematical statement.

Short English statement

The question asks whether an appropriate density threshold can distinguish sampling from interpolation in settings, exemplified by Bergman spaces and wavelet frames, where the source regards Beurling density as inadequate. The spaces, density notion, threshold conditions, and equality case remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003174$, title Critical Sampling Density Beyond Beurling Density, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The record gives proposed year $2016$ and supplies no named proposer. Its statement is marked "exact," with verification described as internal readability and byte-for-byte retention; this does not establish verification against the report.
• Supplied report citation: Learning Theory and Approximation (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/33, pp. 1875--1941. https://doi.org/10.4171/owr/2016/33. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied citation's year $2017$ and the $2016$ appearing in the DOI and proposed-year field are retained without reconciliation.
• Supplied literature reference: Strict density in weighted Fock spaces. https://arxiv.org/abs/1808.02703.
• Supplied literature reference: Critical density for elliptic spectral subspaces. https://arxiv.org/abs/2108.11152.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that critical-density results are available in particular weighted holomorphic and spectral-subspace settings, while the cited context leaves the question open where Beurling density is not the right invariant. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.