(Theoretical project) Theory of Multi-Splines
Splines are piecewise-polynomial functions that satisfy a certain degree of smoothness at their junctions. Along with numerous theoretical properties, splines are also very relevant in practice as they provide a numerically feasible parametrization for continuous functions with a discrete sequence of coefficients. While the theory of splines has been extensively developed in the past 70 years, most published works focus on the case of a single spline space. However, in many applications (\textit{e.g.,} snakes, derivative sampling), it is desirable to consider a sum of spline spaces to increase the flexibility of the model. The goal of this project is to develop the theory of multi-splines, where one assumes that the target (continuously defined) function lies in a sum of multiple spline spaces. The design of optimal generators for these spaces, the development of efficient numerical algorithms, and the study of the fundamental properties (\textit{e.g.,} Riesz basis, approximation power) of these spaces are possible directions in this project.
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