[Math] Constraints on the Fourier transform of a constant modulus function

Question

Considering the function $f:\mathbb{R} \to \mathbb{C}$, with $\left| f(x) \right|=1$ for all $x\in \mathbb{R}$.
Considering $g:\mathbb{R} \to \mathbb{C}$ with $\int_{-\infty}^{\infty}{\left|g(x)\right|^2dx}=1$
I am interested on properties of the amplitude of the Fourier Transform of the product of $f$ and $g$:
$A(k)=\left|FT(f(x)g(x))\right|$
Is there any constraint on $A$ apart from the fact that A is real positive? Considering a fixed $g$, is it possible to attain any $A$ simply by changing $f$?
Thank you for any help

Answer 1

If $g$ happens to be in $L^1$, then the amplitude of the Fourier transform of $fg$ is bounded by the $L^1$ norm of $g$, for any unimodular $f$. This is the only restriction from above since you can always choose $f$ so that $fg\ge 0$, thus bringing the (essential) supremum of $\widehat{fg}$ up to $\|g\|_{L^1}$.

Another part of the question is how small we can make $A$. I guess "arbitrarily small", but don't have a proof. (Except for special case: if $g$ is in $L^1$, then we can chop it into pieces with disjoint supports and small $L^1$ norm, and then use $f$ to move the Fourier transforms of pieces far from one another.)