Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method
Abstract
This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over T-meshes by introducing the concept of TE-connected components, reducing the problem to the rank computation of explicitly constructible conformality matrices. Furthermore, a new class of rectilinear partitions, termed partitions with disjoint truncated l-edges, is introduced. It is proven that under specific conditions, the dimension of the corresponding spline space attains Schumaker's lower bound. This shows that the lower bound is attainable for arbitrary degree d and smoothness order mu in certain partition configurations. Numerical examples, including the Morgan-Scott and Yuan-Stillman partitions, validate the effectiveness and generality of the framework for both triangular and non-triangular rectilinear partitions.
Cite
@article{arxiv.2605.14577,
title = {Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method},
author = {Bingru Huang and Falai Chen},
journal= {arXiv preprint arXiv:2605.14577},
year = {2026}
}