A Theory of Fractional Integration for Generalized Functions

Abstract

In this paper, we develop a theory of fractional integration for certain classes of generalized functions and give one simple application. First, we introduce the appropriate spaces of testing-functions and generalized functions and state some of their basic properties. Next, we discuss the various operators of fractional integration including the Riemann–Liouville and Weyl fractional integrals and the Erdélyi–Kober operators. Use of analytic continuation enables us to obtain a precise description of the mapping properties of these operators relative to the testing-function spaces. We extend the operators to the generalized functions using adjoints and deduce the corresponding mapping properties using standard theorems. Finally, we solve a differential equation involving generalized functions using the previous theory. The theory is much more general than that developed in Erdelyi and McBride [6].