MINIMUM NORM PROPERTIES OF EVEN DEGREE POLYNOMIAL SPLINES WITH RESPECT TO FRACTIONAL DIFFERENTIATION OPERATORS

The existence of minimum norm properties for even degree polynomial splines, analogous to the ones known for odd degree splines, is investigated. It is shown that such properties cannot exist for even degree splines interpolating functions halfway between the partition points. For another class of even degree spline functions, however, which interpolate the local integrals of given functions with respect to the partitions, the sought minimum norm properties can be established in relation to differential operators of fractional degrees. The proof is carried out indirectly by first investigating a generalised problem within the theory of spline systems and then deriving corresponding conclusions.