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 [Appl. Comp. Harmonic Anal.

12 (2002), no. 3, 269–285].

It is a well-known fact that every orthonormal wavelet

φ ∈ L^{2}(

ℝ) with

∣∣ continuous and

(

ξ) =

O(

∣ξ∣^{-1∕2-δ}) as

∣ξ∣→∞ for some

δ > 0 is associated with an MRA. The

ψ constructed in this article is an example of a non-GMRA

C^{∞} frame wavelet with rapid decay. In [Acta Appl. Math.

89 (2005), no. 1-3, 251–270] Baggett et al. gave an example of a non-MRA

C^{r} tight frame wavelet with rapid decay.

[Joint with Marcin Bownik, U. of Oregon]