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Fractional splines and wavelets. (English) Zbl 0940.41004
The authors extend Schoenberg’s family of polynomial splines with uniform knots to all non-integral degrees α>-1. They study two approaches to the construction of the fractional B-splines and show that both approaches are equivalent. They show that the fractional splines share virtually all the properties of the conventional polynomial splines, except that the support of the B-splines for non-integral orders α is no longer compact. They satisfy a two-scale relation and for α>-1/2 they satisfy all the requirements for a multi-resolution analysis of L 2 . As for the usual splines the symmetric fractional splines are solutions of a variational interpolation problem.
MSC:
41A15Spline approximation
41A25Rate of convergence, degree of approximation
65D07Splines (numerical methods)
26A33Fractional derivatives and integrals (real functions)