Recurrence-Coefficient Characterization of Bernstein–Szegő Measures

Definitions and mathematical statement

The requested characterization concerns Bernstein--Szegő measures and the recurrence coefficients of their associated multivariate orthogonal polynomial spaces. The supplied record does not specify the underlying domain, the defining class of measures, the orthogonal spaces, or the recurrence convention. Consequently, the following expresses the intended characterization problem schematically; these missing specifications must be recovered before it becomes a fully determined mathematical statement.

Let $\mathcal B$ denote the class of Bernstein--Szegő measures intended in the cited report. Once the underlying domain, admissible measures, polynomial spaces, basis choices, and recurrence convention have been specified, write $R(\mu)$ for the resulting collection of recurrence coefficients associated with an admissible measure $\mu$.

Characterization problem. Find explicit necessary and sufficient conditions $\mathcal C$ on these recurrence coefficients such that, for every admissible measure $\mu$, \[ \mu\in\mathcal B \quad\Longleftrightarrow\quad R(\mu)\text{ satisfies }\mathcal C. \] The desired characterization is to be analogous to the spectral characterization of Bernstein--Szegő measures on \[ \mathbb T^2=\{(z,w)\in{\mathbb{C}}^2:|z|=|w|=1\} \] mentioned in the source as reference [3]. The supplied record does not identify that reference or specify which features of its characterization should extend to the requested setting.

Unresolved qualifications. The original extraction contains a corrupted formula involving orthogonal spaces, a transformation, and a real span, with symbols including $P_{k,l;\mu}$, $T_{k,l}$, $\omega$, $\widehat p_k$, and $q_2$. Their definitions, index ranges, and the intended equality cannot be recovered reliably from the supplied text. Thus no support, density formula, stability condition, dimensional range, boundary condition, or recurrence normalization is asserted here. The cleaned statement says multivariate, whereas the surviving notation uses two variables; the intended dimensional scope remains unresolved.

Short English statement

Find explicit necessary and sufficient conditions on recurrence coefficients that characterize the Bernstein--Szegő measures intended in the source, analogous to a cited characterization on $\mathbb T^2$. The measure class, orthogonal spaces, and recurrence conventions are missing, so this is a schematic statement requiring recovery of the source definitions and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30005632$, source problem code catalogue entry; supplied complete source record from the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2023.
• Supplied authoritative source citation: Mini-Workshop: Multivariate Orthogonal Polynomials: New synergies with Numerical Analysis (2024), Oberwolfach Reports, pp. 2489--2534, DOI 10.4171/owr/2023/43. Original OWR report: https://doi.org/10.4171/owr/2023/43. The supplied citation labels the workshop 2024, while the DOI and proposed-year metadata indicate 2023; this discrepancy is unresolved.
• Supplied literature source: Geronimo--Benge, bivariate Bernstein--Szegő measures, https://arxiv.org/abs/1102.0986.
• Supplied literature source: Delgado--Geronimo--Iliev--Xu, two-variable class, https://arxiv.org/abs/0710.0035.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports recurrence and parameterization results for bivariate and special multivariate Bernstein--Szegő theories, while describing the requested complete characterization as remaining open. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims; no independent source inspection is claimed here. The original statement's reference [3] is not identified in the supplied record.