Wavelet sets

Prove that if $E$ tiles by translation then $\left\{ e^{2 \pi i k x}\right\}_{k \in \mathbb {Z}}$ is an orthonormal basis for $L^{2}(E)$.

Show that if $E$ tiles both by translation and by dilation and $\widehat{\psi } = \chi_{E}$, then $\left\{ D_{a^{n}} T_{k} \psi \right\}_{k, n \in \mathbb {Z}}$ is an orthonormal basis for $L^{2}(\mathbb {R})$. Therefore $E$ is a wavelet set in this case.

Suppose that $\left\{ D_{a^{n}} T_{k} \psi \right\}_{k, n \in \mathbb {Z}}$ is an orthonormal basis for $L^{2}(\mathbb {R})$, where $\widehat{\psi } = \chi_{E}$. Show that $E$ tiles by translation and by dilation.

Show that $E = \left[-1,-\frac{1}{2}\right] \cup \left[\frac{1}{2}, 1\right]$ is a wavelet set if we take $a = 2$. Verify the explicit formula for the Shannon wavelet $\psi$ given in equation (12.6).

Show that the Journé set $E$ defined in equation (12.65) is a wavelet set (use Exercise 12.4).

Define $\widehat{\psi } = \chi_{E}$, and suppose that $\psi$ was associated with an MRA. Let $\varphi$ be the scaling function for this MRA. Use Exercise 12.38 to show that $\left|\widehat{\varphi }\right| = \chi_{F}$, where

Use the equation $\widehat{\varphi }(\xi ) = m_{0}(\xi / 2) \widehat{\varphi }(\xi / 2)$ to show that $\left|m_{0}(\xi )\right| = 1$ for $\xi \in \left[-\frac{2}{7}, \frac{2}{7}\right]$. Since $m_{0}$ is 1 -periodic, this also shows that $\left|m_{0}(\xi )\right| = 1$ for $\xi \in \left[\frac{5}{7}, \frac{9}{7}\right]$. Show that if $\xi \in \left[1, \frac{8}{7}\right]$ then $\varphi (2 \xi ) = 0$ but $\left|m_{0}(\xi ) \varphi (\xi )\right| = 1$. Conclude that $\psi$ cannot be associated with an MRA.

For an alternative proof that $\psi$ cannot be associated with an MRA, compute $\Phi_{\varphi }(\xi ) = \sum_{k \in \mathbb {Z}}\left|\widehat{\varphi }(\xi +k)\right|^{2}$, and use this to obtain a contradiction.