Random Information for Weighted Approximation Problems

Definitions and mathematical statement

Scope of the supplied problem. Determine the quality of random information for weighted $\ell_p$-balls, with other error norms, and compare it with optimal information. The supplied record does not specify the weights, the range of $p$, the dimension, the approximation operator, the information class, the probability law, or the meaning of quality. Consequently, it does not determine a unique quantified mathematical problem. The following definitions describe a conditional framework, rather than assert missing source hypotheses.

For a chosen finite dimension $d\geq 1$, scalar field $\mathbb K\in\{{\mathbb{R}},{\mathbb{C}}\}$, positive weights $w_1,\ldots,w_d$, and $0<p\leq\infty$, one possible convention for a weighted ball is \[ B_{p,w}^{d}=\begin{cases} \left\{x\in\mathbb K^d:\displaystyle\sum_{j=1}^{d}|w_jx_j|^p\leq 1\right\},&0<p<\infty,\\ \left\{x\in\mathbb K^d:\displaystyle\max_{1\leq j\leq d}|w_jx_j|\leq 1\right\},&p=\infty. \end{cases} \] Here $0<p<1$ gives a quasi-norm ball. This convention and parameter range are illustrative: the source does not select them, and an infinite-dimensional sequence-space interpretation is also unresolved.

More generally, let $B$ be the chosen input ball, let $(Y,\|\cdot\|_Y)$ be the chosen normed error space, and let $S:B\to Y$ be the target map. If information is a map $N:B\to Z_N$, define its optimal worst-case recovery error, when arbitrary recovery maps are permitted, by \[ e(N;B,S,Y)=\inf_{\varphi:Z_N\to Y}\ \sup_{x\in B}\|Sx-\varphi(Nx)\|_Y. \] For a specified nonempty class $\mathcal I_n$ of information maps with budget $n$, the corresponding optimal-information benchmark is \[ e_n^{\mathrm{opt}}(B,S,Y;\mathcal I_n) =\inf_{N\in\mathcal I_n}e(N;B,S,Y). \] A specified probability model $\omega\mapsto N_\omega\in\mathcal I_n$ would then give a random realized error $e(N_\omega;B,S,Y)$. Its distribution or a chosen summary could be compared with this benchmark, subject to appropriate measurability assumptions.

Unresolved target. The source asks for such a determination and comparison for weighted $\ell_p$-balls and other error norms. It supplies neither the parameter regimes nor the random model, admissible recovery rules, error statistic, or intended comparison bounds. In particular, the displayed worst-case formulation is only a possible interpretation; the order of randomization, optimization, and worst-case quantification must be confirmed before a precise source-faithful target can be stated.

Short English statement

The problem asks how well randomly chosen information supports approximation on weighted $\ell_p$-balls, compared with the best permitted information, using other error norms. The supplied record leaves the spaces, weights, random sampling model, information budget, and error criterion unspecified. The definitions above provide a conditional framework requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30004383$, title Random Information for Weighted Approximation Problems, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is named; the supplied proposed year is $2020$.
• Supplied report citation: New Perspectives and Computational Challenges in High Dimensions ($2021$), Oberwolfach Reports, pp. 377--416, DOI 10.4171/owr/2020/6. Supplied source URL and literature-sources link, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2020/6. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked exactly 2026-08-22: the record labels the problem open and states that the quality of random information for other weighted $\ell_p$-balls and other error norms is explicitly identified as open, with no general resolution located. This is supplied dated curation, described as literature triage derived from the supplied Oberwolfach curation; it has not been independently verified here. Time-sensitive status and cited claims require independent verification.
• The supplied statement-status label is corrected_verified. Its accompanying note says that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. This is a supplied editorial assertion, not independent mathematical verification of this draft.