Fractional integrals of generalised functions

Abstract

The concept of integrals of fractional order of a function f, defined by if Reα > 0, can be extended to generalised functions in the framework of the theory of convolution of distributions. The resulting theory [2, Chap. I §5.5] is very satisfactory for many purposes but there are circumstances in which it is not suitable. Such circumstances arise in particular if it is necessary to multiply, before or after integratrion, by non-integral powers of the variable. Pointwise multiplication by fractional powers of the independent variable does not make sense in the theory of distributions.