Can I express the following double summation
$$\sum_{(i,j)\in\mathcal{R}} A_{v_i} G(v_j-v_i)$$
where $\mathcal{R}=\{ (i,j) \in \mathbb{Z}^2,i \in [1:n], j \in [1:m]\}$ while $G(.)$ is any function of (.) and $A_{v_{i}}$ is any function of $v_{i}$ as
$$\sum_{i=1}^n A_{v_i} D_{v_i}$$
where $D_{v_i} = \sum_{j=1}^m G(v_j-v_i)$.
Does anyone know if the simplification is right? I basically would like to have the expression in terms of $i$ only.
Can anyone think of another way?