The scope of a functional calculus approach to fractional differential equations

This paper presents a functional calculus definition of linear fractional (pseudo)differential operators via generalised Fourier transforms as a natural extension of integer ordered derivatives. First, we describe the extension of our L-2-based functional calculus approach on D'. Second, we demonstrate that important computational rules as well as properties of integer ordered differential operators are preserved by our approach. This concerns also the D'-kernel of our operators which belong to the same class as those corresponding to integer derivatives, i.e., they are linear combinations of "polynomials times exponential functions".