Abstract:
The transformation of cubic Schoenberg splines is carried out using four auxiliary cubic Schoenberg splines having finite supports, whose sizes are smaller than the size of the mother spline’s finite support. As a result, eight grid sets of orthogonal cubic real-valued Schoenberg splines are constructed. The approximation properties of these splines are investigated. It is shown that the approximation order of Schoenberg splines, that are also modified by Schoenberg splines, is significantly higher than the approximation order of step function-modified Schoenberg splines, and coincides with the approximation order of classical cubic Schoenberg splines. The defect of modified Schoenberg spline is equal to one, as that of classical Schoenberg spline. The modified spline is a continuous function which has no breaks in the first and the second derivatives at the points where the parts of the mother spline and the parts of the splines used for modification meet.
Keywords:cubic Schoenberg splines, orthogonalization of compactly supported functions, spline approximation
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