I'm trying to wrap my head around the notion of translation invariance in terms of dictionaries/wavelets. For example in Lecture Notes, Page 41 its written that one starts with a family of atoms/wavelets $(\psi_j)_{j\in \mathbb{Z}}$ and adds all translates $t\in \mathbb{R}$ via defining \begin{equation*} \psi_{j,t}(x) := \psi_j (x - t). \end{equation*} Now the calculation of the coefficients for a singal $f$ becomes \begin{equation}\label{calc coeff} \Phi f (j,t) = \left\langle f, \psi_{j,t}\right\rangle = \int_\mathbb{R} f(x) \psi_j^\ast (t - x) \mathrm{d}x = f \ast \psi_j^\ast (t), \end{equation} where $\psi_j^\ast(x) := \psi_j(-x)$. To me, translation invariance means something like $f(x) = f(x-t)$, but I don't understand/see where this equations holds for the above "argumentation"?
Furthermore, reading some Wikipedia on Stationary Wavelet Transform, it says in the first sentence "to overcome the lack of translation-invariance of the discrete wavelet transform", i.e. when the translation $t\in \mathbb{R}$ becomes $t\in \mathbb{Z}$. Why is the discrete wavelet transform not translation invariant? The calculation for the coefficients $\left\langle f, \psi_{j,t} \right\rangle$ is the same, but only discrete convolution instead of continuous?