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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 $$\alpha>-1$$. 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 $$\alpha$$ is no longer compact. They satisfy a two-scale relation and for $$\alpha>-1/2$$ 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:
 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 65D07 Numerical computation using splines 26A33 Fractional derivatives and integrals
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