$$ \newcommand{\pdot}{(\cdot)} \newcommand{\pnum}[1]{{(#1)}} \bm{f}^\pnum{j}(\cdot) = \hat{\bm{f}}^\pnum{j}\pdot + \bm{f}^\pnum{j+1}\pdot $$ where $\bm{f}^\pnum{j+1}\pdot$ is the remainder process at resolution $j+1$ whose noisy instantiations on $\mathcal{T}^\pnum{j+1}$ are modeled according to, $\forall \bm{x}_t \in \bm{x}_{\mathcal{T}_l^{(j+1)}}$: $$ \bm{z}_{t,l}^\pnum{j+1} = \sum_{i=1}^p a_{i,l}^\pnum{j+1} \bm{u}_i \phi_i^\pnum{j+1}(\bm{x}_t, \bm{\tau}_l^\pnum{j+1}) + \bm{b}_l^\pnum(j+1) + \bm{e}_{t,l}^\pnum{j+1} $$

The recursive procedure continues until resolution $j = m$ is reached. By assuming that the latent remainder process at $j = m + 1$ approaches zero, we can approximate $f\pdot$ as the sum of the predictive processes from all resolutions, $$ \bm{f}\pdot = \bm{f}^\pnum{m+1}\pdot + \sum_{j=0}^m \hat{\bm{f}}^\pnum{j}\pdot \approx \sum_{j=0}^m \hat{\bm{f}}^\pnum{j}\pdot $$