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

A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces