Splines and fractional differential operators

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form Ly = 0 where L is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.