@article{Xu2026, 
author = {Xiaoda Xu},
title = {The partition principle revisited: Non-equal volume designs achieve minimal expected approximation error in function sampling},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {15448-15468},
keywords = {random sampling, stratified sampling, function approximation, star discrepancy, partition principle},
url = {https://www.sciopen.com/article/10.3934/math.2026635},
doi = {10.3934/math.2026635},
abstract = {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:         E    ‖  f  −            A        Z    f  ‖  <      E    ‖  f  −            A        Y    f  ‖  ,where    Z and    Y denoted random samples drawn from the non-uniform-volume and jittered designs, respectively, and        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        O    (      N          −      1              /            2      −      1              /            (      2      d      )        ) 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.}
}