  Here we give a new approach to the Paley--Wiener theorem in a Mellin analysis
setting which avoids the use of the Riemann surface of the logarithm and
analytical branches and is based on new concepts of "polar-analytic function"
in the Mellin setting and Mellin--Bernstein spaces. A notion of Hardy spaces in
the Mellin setting is also given along with applications to exponential
sampling formulas of optical physics.
