Does absolute or conditional convergence of the integrals define fractional derivatives?

Abstract

In this master’s thesis we analyse the class of fractionally differentiable functions. This work is built on Gennadi Vainikko’s recent paper “Which functions are fractionally differentiable?”, that characterises the class of fractionally differentiable functions in terms of the pointwise convergence or equiconvergence of certain improper integrals containing these functions. The aim of this thesis is to present and analyse an example, which shows us that in order to obtain all fractionally differentiable functions, one may not replace the conditional convergence of certain integrals by their absolute convergence. Also some supporting lemmas are formulated and proved.