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A spline is a smooth piecewise polynomial function defined on a subdivision of R^n. In this project, we study the module of all splines over R and R^2. Our goal is to determine when this module is free and to find an explicit module basis. The subset of all splines with degree less than or equal to k forms a vector space over R, and we find the dimension of this vector space when n=1. We then consider splines with boundary conditions and show that this is a module as well. We try to determine when this module is free, and we prove a general result about its Hilbert Series.
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Scoppetta, Lindsey, "Modules of Splines with Boundary Conditions" (2012). Senior Projects Spring 2012. 81.