Fractional splines and wavelets. (English) Zbl 0940.41004

Unser, Michael; Blu, Thierry

SIAM Rev. 42, No. 1, 43-67 (2000).

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.

Reviewer: Ganesh Datta Dikshit (Auckland)

MSC:

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

Keywords:

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