A new bound for smooth spline spaces
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For a planar simplicial complex Delta contained in R^2, Alfeld-Schumaker proved that 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) when k >= 3r+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.
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