Closed Forms for Multivariate Polynomial Approximation Kernels

Definitions and mathematical statement

A coefficient of a polynomial is understood relative to a specified polynomial basis: schematically, if $p_1,\ldots,p_M$ are basis polynomials, then a polynomial kernel can be written as \[ K=\sum_{j=1}^{M}c_jp_j. \] Here this expression only explains the meaning of coefficients; the supplied record does not specify the variables, scalar field, basis, admissible kernels, or optimization problem defining the intended kernel.

Problem (as identified by the supplied record). Can the coefficients of the optimal multivariate polynomial-approximation kernel referred to in the cited Oberwolfach report be expressed in closed form?

Unresolved formulation. The supplied excerpt states only that finding a closed form "for the multivariate case as well" is an open question. It does not define the approximation domain, target functions, kernel construction, degree parameters, admissibility conditions, objective defining optimality, or meaning of closed form. It also does not specify whether an optimizer is unique or whether the question concerns some or every optimal kernel. Consequently, the intended parameter ranges, boundary conditions, and quantifiers cannot be recovered from this record. The displayed expansion is schematic and does not supply these missing hypotheses. A precise quantified formulation requires the relevant definitions and surrounding passage from the cited report.

Short English statement

The supplied problem asks for closed-form coefficients of an optimal multivariate polynomial-approximation kernel. The kernel and its optimization criterion are not defined in the supplied excerpt, so the exact mathematical target remains unresolved. The record labels the problem open; that label is supplied curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005089$, title Closed Forms for Multivariate Polynomial Approximation Kernels, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2022 and names no proposer.
• Supplied report citation: Conic Linear Optimization for Computer-Assisted Proofs (2023), Oberwolfach Reports, pp. 1039--1089, DOI 10.4171/owr/2022/20. https://doi.org/10.4171/owr/2022/20. The literature-source list labels this reference "Oberwolfach Report 20/2022." The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: Kirschner--de Klerk, multivariate approximation kernels. https://arxiv.org/abs/2203.05892.
• Literature assessment dated 2026-08-22: the supplied curation labels the question open and reports that subsequent work gives constructions without resolving the multivariate closed-form question. This is supplied dated curation, not independently verified here. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry assigns statement status corrected_verified and reports that its clean statement was checked against the official report and preserves its hypotheses, notation, and target. This is a supplied verification claim; the excerpt provided here does not contain the definitions needed to check it independently.