Stable Oversampling Approximations of Linear Systems

Definitions and mathematical statement

For specified signal vector spaces $X$ and $Y$ over a common scalar field $K$, let $S_\tau^X:X\to X$ and $S_\tau^Y:Y\to Y$ denote compatible time translations for each permitted translation parameter $\tau$. A map $T:X\to Y$ is linear and time-invariant if \[ T(af+bg)=aT(f)+bT(g) \quad\text{for every }a,b\in K\text{ and }f,g\in X, \] and \[ T S_\tau^X=S_\tau^Y T \quad\text{for every permitted }\tau. \] This gives only the algebraic meaning of linear time-invariance; the supplied record does not specify $X$, $Y$, $K$, or the permitted translations, and does not state whether boundedness is required.

Source question. Characterize the linear time-invariant systems that can be stably approximated by approximation processes using oversampling.

Unresolved formulation. The supplied record does not define the signal class, measurements, approximation processes, sampling threshold relative to which sampling is excessive, or the meaning of stability. It also does not specify a convergence topology, the variables in which convergence must hold uniformly, or the dependence of any stability bound on the input, sampling rate, or approximation index. Consequently, the question above is a characterization request whose precise mathematical predicate remains unspecified. The displayed algebraic definitions do not resolve these omissions or select a particular sampling model.

Short English statement

The question asks which linear time-invariant systems admit stable approximation using oversampling. A precise characterization requires the intended signal spaces, sampling model, approximation procedures, and definitions of stability and convergence. These are absent from the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002094$, title Stable Oversampling Approximations of Linear Systems, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2012$ and names no proposer.
• Supplied report citation: Applied Harmonic Analysis and Sparse Approximation (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/29, pp. 1759--1843. Source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2012/29. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Signal and system approximation from general measurements, https://arxiv.org/abs/1402.1092. No authors or more detailed bibliographic information are supplied in the record.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and reports that later work provides stable approximation under broader measurement models and selected sampling schemes, while a general characterization remains unresolved. This is supplied dated curation and literature triage, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.