Classical and Approximate Exponential Sampling Formula: Their Interconnections in Uniform and Mellin-Lebesgue Norms

We show that the exponential sampling theorem and its approximate version for functions belonging to a Mellin inversion class are equivalent in the sense that, within the setting of Mellin analysis, each can be obtained from the other as a corollary. The approximate version is considered for both, convergence in the uniform norm and in the Mellin--Lebesgue norm. An important tool is the introduction of a Mellin version of the mixed Hilbert transform and its continuity properties. Our paper extends the analogous equivalence between the classical and the approximate sampling theorem of Fourier analysis.