A necessary and sufficient condition for dual Weyl-Heisenberg frames to becompactly supported

In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expa nsions with rational oversampling. We present a necessary and sufficient co ndition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L-2(R) for its minimal dual (Wexler-Raz dual) gamma(0)(t) to be compactly supported. We furthermore provide a necessary and sufficient cond ition for a band-limited function g(t) generating a Weyl-Heisenberg frame f or L-2(R) to have a band-limited minimal dual gamma(0)(t). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t) has a compactl y supported (band-limited) minimal dual gamma(0)(t) if and only if the Weyl -Heisenberg frame operator is a multiplication operator in the time (freque ncy) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi re presentation of the Weyl-Heisenberg frame operator, and on the theory of po lynomial matrices.