Space-dependent estimates for the admissibility threshold

Establish space-dependent estimates for the threshold parameter J under the minimal admissibility hypotheses used for randomized least-squares quadrature construction.

Background

The paper develops randomized weighted least-squares procedures for constructing exact and stable quadrature rules, with finite-sample guarantees governed by relative admissibility. The admissibility assumptions ensure that the sample-size parameter J can be chosen sufficiently large, but the resulting choice is generally implicit and may depend on tail behavior associated with the sampling measure, the Christoffel-type function, and the Riesz representer.

The unresolved problem is to identify estimates for J that explicitly reflect the spatial structure of the domain and the relevant functions while requiring only the paper’s minimal admissibility conditions. Such estimates would make the sample-complexity guarantees more transparent and quantitatively useful in general settings.