Dirichlet Splines as Fractional Integrals of $B$-Splines

Abstract

Using Dirichlet averages we generalize the notion of a classical divided difference of a function by introducing a parameter r in R k+1 + . The case r in N k+1 is related to divided differences with multiple knots. We give an interpretation of these generalized differences in terms of fractional operators applied to classical divided differences considered as functions of their knots. The result is then applied to show that Dirichlet splines can be seen as fractional derivatives of B-splines.