Salem conditions in the non-periodic case

In the classical sources, Salem's necessary conditions for a trigonometric series to be the Fourier 
series of an integrable function are given in terms of ``some" sums. Realizing that, in fact, they 
are given in terms of the discrete Hilbert transforms, we generalize these to the non-periodic case, 
for functions from the Wiener algebra. Other relations of the two objects are also discussed.
The obtained necessary condition is used to construct a monotone function with non-integrable cosine 
Fourier transform in a much easier way than in the classical book \cite{Ti} by Titchmarsh.
Certain open problems are posed.