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Talks given at NuHAG events

The Canonical and Alternate Duals of a Wavelet Frame


  Jakob Lemvig

  given at  strobl07 (17.06.07)
  id:  608
  length:  min
  status:  accepted
  type:  poster
  www:  http://www.mat.dtu.dk/English/Medarbejdere/MAT_PHD.aspx?lg=showcommon&id=21544&type=person
  LINK-Preprint:  http://www2.mat.dtu.dk/people/J.Lemvig/cadwf.pdf
  LINK-Presentation: 
  ABSTRACT:
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() with ˆφcontinuous and ˆφ(ξ) = O(ξ-12-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]


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