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!