Integral transforms based upon fractional integration

Abstract

AbstractThe theory of Fourier transformscan be developed from the functional equation K(s) K(1 – s) = 1, where K(s) is the Mellin transform of the kernel k(x).In this paper I show that reciprocities can be obtained which are analogous to the Fourier transforms above but which develop from the much more general functional equationThe reciprocities are obtained by using fractional integration. In addition to the reciprocities we have analogues of the Parseval theorem and of the discontinuous integrals usually associated with Fourier transforms.In order to simplify the analysis I confine myself to the case n = 1 and to L2 space.