An algebraic definition of fractional differentiation

Abstract

Suppose I is an open real interval, and V a subspace of C^∞(I) invariant under D, differentiation. It is shown that if V contains a polynomial of positive degree, then D has no homogeneous q^th root on V, for any integer q > 1. This conclusion is not generally true if I is replaced by an arbitrary open set in the reals. Necessary and sufficient conditions for the existence of q^th roots of D on finite dimensional subspaces of C^∞(I) invariant under D, in terms of the differential equations of which such subspaces are the solution spaces, are obtained.