Which Functions are Fractionally Differentiable

s of MMA2015, May 26–29, 2015, Sigulda, Latvia c © 2015 WHICH FUNCTIONS ARE FRACTIONALLY DIFFERENTIABLE? G. VAINIKKO Institute of Mathematics, University of Tartu Liivi 2, Tartu 50409, Estonia E-mail: gennadi.vainikko@ut.ee We define a fractional differentiation operator as the inverse to Riemann-Liouville integral operator, and examine the relations of this most natural concept with more popular fractional differentiation operators of Riemann-Liouville and Caputo. Our main result concerns the description of the range of Riemann-Liouville integral operator in the space of continuous functions. As the result we can describe, in particular, the class of functions that are differentiable in the sense of Riemann-Liouville and Caputo. Also the Abel equation with coefficient function of two variables can be examined on the basis of Riemann-Liouville’s operator inversion.