In a functional space like $L^{2}[0,1]$ with some inner product, we may wish to find an orthogonal basis $\{ \phi_{i} \}$ whose linear combinations are dense in the space. Wavelets have finite time intervals as support, so they provide time and frequency localization: the original function is decomposed into local patterns and fixed frequencies. ![[Wavelets.png#invert|center|w90]] The symmlet-8 wavelets sacrifices some localization for smoother bases. A family of wavelet bases a generated by **fathers** $\phi(t)$. They are of the form $\phi_{j,k}(t):=2^{j / 2}\phi(2^{j}t-k)$which shrinks the support by a scale of $2^{j}$ and translates it by $k \in \mathbb{Z}$. The **reference space** is spanned by the translations of the father: $V_{0}:=\mathrm{span}\{ \phi_{0,k}(t)=\phi(t-k)\,|\, k \in \mathbb{Z} \}$In general, there is $\begin{align*} V_{j} :&= \mathrm{span}\{ \phi_{j,k}(t)\,|\, k \in \mathbb{Z} \}\\[0.8em] \cdots&\supset V_{j+1} \supset V_{j} \supset V_{j-1} \supset \cdots \end{align*}$where *$V_{j}$ with larger $j$ contains higher frequency wavelets, hence more detailed representations*.