A DIRECT APPROACH TO THE MELLIN TRANSFORM

The aim of this paper is to present an approach to the Mellin transfor m that is fully independent of Laplace or Fourier transform theory, in a systematic, unified form, containing the basic properties and major results under natural, minimal hypotheses upon the functions in quest ions. Cornerstones of the approach are two definitions of the transfor m, a local and global Mellin transform, the Mellin translation and con volution structure, in particular approximation-theoretical methods co nnected with the Mellin convolution singular integral enabling one to establish the Mellin inversion theory. Of special interest are the Mel lin operators of differentiation and integration, more correctly of an ti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These t wo operators are different than those considered thus far and more gen eral. They are of particular importance in solving differential and in tegral equations. As applications, the wave equation on R+ x R+ and th e heat equation in a semi-infinite rod are considered in detail. The p aper is written in part from an historical, survey-type perspective.