Mellin-Fourier series and the classical Mellin transform

The aim of this paper is to present the counterpart of the theory of Fourier series in the Mellin setting, thus to consider a finite Mellin transform, or Mellin-Fourier coefficients, together with the associated Mellin-Fourier series. The presentation, in a systematic and overview form, is independent of the Fourier theory (or Laplace transform theory) and follows under natural and minimal assumptions upon the functions in question. This material is put into connection with classical Mellin transform theory on + via the Mellin-Poisson summation formula, also in the form of two tables, as well as with Fourier transform theory. A highlight is an application to a new Kramer-type form of the exponential sampling theory of signal analysis.