Fourier Coefficients of Beurling Functions and a Class of Mellin Transform Formally Determined by its Values on the Even Integers

n∈Z |c(n)| 2 is a possible tool to compute or estimate this norm. In this note we give an expression for the Fourier coefficients c(n) of f+1, when f is a function defined as above. As an application, we derive an expression for Mf (s) := ∫ 1 0 (f(x)+1)x s−1 dx as a series that only depends on Mf (2k), k ∈ N. We remark that the Fourier coefficients c(n) depend on Mf (2k) which, for a function f defined as above, can be expressed also in terms of the ak’s and θk’s. Therefore, a better control on these parameters will allow to estimate Mf (2k) and therefore eventually to handle ‖f + 1‖ via our expression for the Fourier coefficients and Parsevall Identity.