A new bound for smooth spline spaces

TL;DR

This paper establishes a new upper bound on the polynomial degree for which the dimension of planar spline spaces equals a known polynomial, refining previous bounds and clarifying when equality holds.

Contribution

The authors prove that the dimension equality between spline space and its lower bound polynomial fails for degrees below a specific threshold, improving understanding of spline space dimensions.

Findings

Dimension equality fails for k <= (22r+7)/10

Previous bounds are refined with a new, sharper threshold

Provides conditions under which the polynomial bound is tight

Abstract

For a planar simplicial complex Delta contained in R^2, Schumaker proved that a lower bound on the dimension of the space C^r_k(Delta) of planar splines of smoothness r and polynomial degree at most k on Delta is given by a polynomial P_Delta(r,k), and Alfeld-Schumaker showed this polynomial gives the correct dimension when k >= 4r+1. Examples due to Morgan-Scott, Tohaneanu, and Yuan show that the equality dim C^r_k(Delta) = P_Delta(r,k) can fail when k = 2r or 2r+1. We prove that the equality dim C^r_k(Delta)= P_Delta(r,k) cannot hold in general for k <= (22r+7)/10.