The canonical and alternate duals of a wavelet frame

We show that there exists a frame wavelet $\psi$ with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. On the other hand, there exists infinitely many alternate duals of $\psi$ generated by a single function. Our argument closes a gap in the original proof of this fact by Daubechies and Han [Appl. Comp. Harmonic Anal. {\bf 12} (2002), no.~3, 269-285].