My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in Fourier domain by $$ \widehat{\psi}_{\mathrm{Shannon}} (\omega) = \chi_{[\pi, 2\pi]}(\omega) + \chi_{[- 2 \pi, - \pi]}(\omega)$$ where $\chi_A$ is the indicator function of $A$.

Some authors are claiming that "Daubechies wavelets converge to the Shannon wavelet". But I could not find in which sense this has to be understood. Do we have the following convergences:

  1. $\psi_n$ converges uniformly to $\psi_{\mathrm{Shannon}}$ on any compact set $K \subset \mathbb{R}$?
  2. $\widehat{\psi}_n$ converges pointwise to $\widehat{\psi}_{\mathrm{Shannon}}$?

  3. $\lVert \psi_n - \psi_{\mathrm{Shannon}} \rVert_p \underset{n\rightarrow \infty}{\longrightarrow} 0$, for $1\leq p \leq \infty$?

  • $\begingroup$ Interesting - do you have a reference for the claim? Well 2. can not hold, since the Fourier transform of the Shannon wavelet is discontinuous while the transforms of the Daubechies wavelet are continuous (if I remember correctly). $\endgroup$ – Dirk Sep 22 '15 at 20:07
  • $\begingroup$ Well, after your edit, 2 can well be. It would be good to make edits in a way that comments and answers (if there are any) still make sense… $\endgroup$ – Dirk Sep 23 '15 at 11:53
  • $\begingroup$ The convergence of Daubechies wavelets to Shannon when the order increases to infinity seems to be part of the folklore of the community of mathematical signal processing. It is moreover probably true for other families of wavelets. This claim can be found for instance without reference in link.springer.com/chapter/10.1007%2F978-3-319-12577-0_51 . Interesting results were obtain for the Battle-Lemarié wavelets in mathsci.kaist.ac.kr/bk21/morgue/research_report_pdf/05-1.pdf . You are right for the compact convergence in Fourier. I edited my question accordingly. $\endgroup$ – Goulifet Sep 23 '15 at 11:58
  • $\begingroup$ I should also mention the following paper, but it is not clear to me how the results are related with my question: sciencedirect.com/science/article/pii/S1063520397902347 $\endgroup$ – Goulifet Sep 23 '15 at 11:59

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