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:

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

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