Suppose that $f(t)$ is a square-integrable, band-limited function, i.e. the Fourier transform $\hat f$ has compact support.


**Problem:** Under which assumptions on a function $g(t)$ is the map $h(t) := e^{ig(t)}f(t)$ band-limited, i.e. $\hat h$ has compact support.


My idea would be to use the Paley-Wiener theorem which says $f$ extends to an entire function of exponential type (in particular, it has order less or equal than 1). Now one has to find assumptions on $g$ so that $h$ is again of exponential type (maybe $g$ must be a linear function then?).


Thanks in advance for any help!



Under which assumptions is $e^{ig(t)}f(t)$ of exponential type if $f$ is of exponential type?
A function is entire and of exponential type if and only if its Fourier transform has bounded support (This is Paley-Wiener theorem). Since your
$f$ belongs to this class, then, assuming that $g$ is also entire, $h=e^gf$ will belong to this class if and only if $g$ is linear.


If you do not want to assume that $g$ is entire, since $h$ and $f$ are entire, $e^g$ is meromorphic, with poles contained in the zero set of $f$,
and of exponential type in the sense that $T(r,e^g)=O(r),$ where $T$ is the Nevanlinna characteristic. This is a complete characterization: take any meromorphic function $F$ of exponential type whose poles belong to the zero set of $f$, and take $g=\log F$.


