A self-contained approach to Mellin transform analysis for square integrable functions; Applications

In several papers the authors introduced a self-contained approach (independent of Fourier or Laplace transform theory) to classical Mellin transform theory as well as to a new finite Mellin transform in case the functions in question are absolutely (Lebesgue) integrable. In this paper the matter is considered for functions which are basically square-integrable. The unified and systematic approach presented, which is valid under minimal and natural hypotheses, is applied to sampling analysis, particularly to that connected with Kramer's lemma. An application is a clean approach to exponential sampling theory (of optical circles) for signals which possess certain integral representations, but also for Mellin-bandlimited signals as well as for those which are only approximately Mellin-bandlimited.