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Zbl 0940.41004
Unser, Michael; Blu, Thierry
Fractional splines and wavelets. (English)
[J] SIAM Rev. 42, No.1, 43-67 (2000). ISSN 0036-1445; ISSN 1095-7200/e

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.
[Ganesh Datta Dikshit (Auckland)]

MSC 2000:: *41A15 Spline approximation
41A25 Degree of approximation, etc.
65D07 Splines (numerical methods)
26A33 Fractional derivatives and integrals (real functions)

Keywords: fractional derivatives; Riesz basis; B-splines; variational interpolation problem

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