This paper investigated the expected approximation error in function recovery via a novel class of non-uniform-volume partitions. We established two main theoretical results. First, we proved a strong partition principle showing that stratified sampling based on our proposed non-uniform-volume partitions yielded a strictly smaller expected approximation error than classical jittered sampling:

$$\mathbb{E}\Vert f-{\mathcal{A}}_{Z}f\Vert <\mathbb{E}\Vert f-{\mathcal{A}}_{Y}f\Vert ,$$

where $Z$ and $Y$ denoted random samples drawn from the non-uniform-volume and jittered designs, respectively, and $\mathcal{A}$ denoted the piecewise-constant approximation operator. Second, we derived explicit, dimension-explicit upper bounds on the expected approximation error under our non-uniform-volume partition framework—bounds that improved upon the best-known rates for jittered sampling at the constant level. We wish to emphasize that the improvement was at the constant level only: the asymptotic convergence rate $\mathcal{O}(N^{-1/2-1/(2d)})$ remained unchanged from classical jittered sampling. Nevertheless, we believed that constant-level improvements can be practically significant and theoretically illuminating. Collectively, these results offered a theoretical basis for the use of non-uniform-volume partitions in high-dimensional function approximation and sampling theory.

This paper is dedicated to the estimation of the probabilistic upper bounds of star discrepancy for Hilbert's space filling curve (HSFC) sampling. The primary concept revolves around the stratified random sampling method, with the relaxation of the stringent requirement for a sampling number $N=m^d$ in jittered sampling. We leverage the benefits of this sampling method to achieve superior results compared to Monte Carlo (MC) sampling. We also provide applications of the main result, which pertain to weighted star discrepancy, $L_2$-discrepancy, integration approximation in certain function spaces and examples in finance.