Systematic construction of graphon sampling partitions

Construct the partitions \(\{S_1,\ldots,S_k\}\) required for the graphon Paley–Wiener sampling approach, ideally with control over the associated constants entering the sampling estimate.

Background

The sampling estimates developed in the paper require a measurable partition $\{S_1,\ldots,S_k\}$ of $[0,1]$ and sampling functions supported on the partition elements. The constants in the estimates depend on the partition and sampling functions, including the quantities controlling the spectral gap and normalization of the sampling functionals.

The paper does not provide a systematic method for constructing such partitions. It identifies as an unresolved problem the development of a construction procedure that also controls the constants appearing in the sampling estimate, which would make the graphon sampling framework more effective and potentially improve its consistency and robustness properties.