I came across the following assertion: if $f\in PW\_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,a]$, then $$\int\_0^{+\infty}\frac{\ln\frac{1}{|f(x)|}}{1+x^2}\mathrm{dx}<+\infty.$$ The authors say it's a classical result but I can't seem to find the proof or come with an idea to prove it. Any suggestions/references?



Integral of $\ln(1/|f|)$ for $f$ bandlimited
This is due to Wiener and Paley, (but can be also derived from the Jensen inequality and description of positive harmonic functions in
a half-plane) and a more general formulation is this:
If $f$ is entire, of exponential type, and
$$\int\frac{\log^+|f(x)|}{1+x^2}dx<\infty,$$
then
$$\int\frac{\log^-|f(x)|}{1+x^2}dx<\infty.$$
Here $a^+=\max\{ a,0\}$ and $a^-=(-a)^+$, so that $a=a^+-a^-$.


There is a whole book about these integrals:


P. Koosis, The logarithmic integral, 2 vols. Cambridge Univ. Press, 1988, 1992. (The fact that I stated is proved in section III G 2
of volume I.)


