Finite-Dimensional Approximation of Bounded Analytic Functions

Definitions and mathematical statement

Question. Let $f_1,\ldots,f_k\in H^\infty(D,{\mathbb{C}}^N)$ and $M(z)=\operatorname{span}_{{\mathbb{C}}}\{f_1(z),\ldots,f_k(z)\}$. For each $\varepsilon>0$, does there exist a complex linear subspace $S\subset{\mathbb{C}}^N$, with dimension bounded only in terms of $k,\varepsilon$, such that every admissible bounded analytic function $g$ satisfies $\operatorname{dist}_{\ell_p}(g(0),S)\le\psi(\varepsilon)$, for a fixed function $\psi$ tending to zero as $\varepsilon\downarrow0$? Here admissible means\[g\in H^\infty(D,{\mathbb{C}}^N),\qquad \|g(z)\|_p\le1\quad(z\in D),\qquad \operatorname{dist}_{\ell_p}(g(\zeta),M(\zeta))\le\varepsilon\quad\text{for a.e. }\zeta\in\partial D.\]

Using the usual unit-disc interpretation $D=\{z\in{\mathbb{C}}:|z|<1\}$, $H^\infty(D,{\mathbb{C}}^N)$ denotes bounded holomorphic maps. Boundary values are their almost-everywhere nontangential limits with respect to arc length, and $M(\zeta)$ is the span of the corresponding boundary values of the $f_j$. For a fixed intended exponent $p$, distance means $\operatorname{dist}_{\ell_p}(v,S)=\inf_{s\in S}\|v-s\|_p$; the source does not supply the allowed range of $p$. The $f_j$ are bounded, but the record imposes no common unit bound on them.

The essential uniformity is that a bound $d(k,\varepsilon)$ for $\dim S$ is independent of the ambient dimension $N$ and of the chosen functions. For each given $N,k,(f_j),\varepsilon$, the subspace $S$ may depend on these data but must be chosen before $g$ and work simultaneously for all admissible $g$. The error modulus must be uniform in $N,(f_j),g$ with $\psi(\varepsilon)\to0$; whether it may additionally depend on $k$ or the fixed $p$ is not stated. Likewise, dependence of the dimension bound on $p$, if a range of $p$ is intended, is not specified. Taking $S$ separately for each $g$, or allowing its dimension bound to depend on $N$, would change the question.

The supplied literature triage checked 2026-08-21 labels the question open and reports positive answers for variants, but no located solution of the full report formulation. Those variants and their hypotheses are not supplied, and this dated assessment requires independent verification.

Short English statement

A family of pointwise spans of $k$ bounded analytic vector functions nearly contains the boundary values of every admissible $g$. The question asks whether all corresponding center values lie close to one subspace whose dimension is bounded independently of $N$. That subspace must work for all such $g$ at once.

Sources

• Local UnsolvedMath registry, permanent ID 30000117, code catalogue entry; Finite-Dimensional Approximation of Bounded Analytic Functions, Oberwolfach Reports Open Problems.
• Compactness Problems in Interpolation Theory and Function Spaces (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/40, pp. 2103--2132. https://doi.org/10.4171/owr/2004/40. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label OWR report: https://doi.org/10.4171/owr/2004/40.
• Supplied literature label Report PDF: https://oa.tib.eu/renate/bitstreams/b5a34840-008a-45dd-92d4-721151bffbc2/download. This is a supplied reference only; it was not downloaded.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports source inspection, not independently checked here.