The Canonical and Alternate Duals of a Wavelet Frame
given at strobl07 (17.06.07)
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 ∈ L2(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 Cr tight frame wavelet with rapid decay.
[Joint with Marcin Bownik, U. of Oregon]