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.


41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
65D07 Numerical computation using splines
26A33 Fractional derivatives and integrals
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