The canonical and alternate duals of a wavelet frame

摘要

We show that there exists a frame wavelet Ψ 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 Ψ generated by a single function. Our argument closes a gap in the original proof of this fact by Daubechies and Han [The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (3) (2002) 269-285].