2026/07/25 by Alexander Panchenko, Ben Hellwig, Oleh Rudenko
#math.NA #cs.NA #math.CA #math.CO #math.MG
We develop a mesoscopic framework for analyzing perturbations of finite point sets. Given a reference node set Y with known cubature and approximation properties, we consider a disordered node set Q that is observed only through its populations in cubes at scale r>0. By imposing Hall-type (HT) combinatorial constraints on these populations, we prove the existence of a perfect matching between Y and Q with O(r) range. This allows integral approximation estimates on coarser cubes at scale h>r to be transferred from Y to Q with explicit error control and anchors Q to a periodic grid. We then use translation-invariant quasi-interpolation methods to obtain high-order estimates of order hs as in the quasi-uniform setting, but for a different class of geometries. The key restrictions are the HT conditions and the bound r≤ Ch, where C<1 is scale independent.