Prescribed correction supports

Let

The closure of $N$ in $W$ is contained in $\mathcal{S}$.

The class ${\mathfrak{C}}_S$ consists of all distribution-valued families $c_{\delta }$ with $\mathrm{supp}{c}_{\delta }\subset \mathcal{S}$ at every scale, without order or scale restrictions. Define ${\mathfrak{C}}_N$ and ${\mathfrak{C}}_L$ similarly using $\overline{N}$ and $\mathcal{L}$. These supports are fixed before coefficient collection.