Multivariate discrete splines and linear Diophantine equations

Jia, Rong Qing

doi:10.1090/S0002-9947-1993-1159194-6

Multivariate discrete splines and linear Diophantine equations

Author: Rong Qing Jia
Journal: Trans. Amer. Math. Soc. 340 (1993), 179-198
MSC: Primary 41A15; Secondary 11D04, 39A10, 39A70, 41A63
DOI: https://doi.org/10.1090/S0002-9947-1993-1159194-6
MathSciNet review: 1159194
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Abstract: In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly independent. In order to understand the local structure of discrete splines we develop a general theory for certain systems of linear partial difference equations. Using this theory we prove that the integer translates of a discrete box spline are locally linearly independent if and only if they are linearly independent.

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