Let $A$ be an orthogonal matrix. Then the proximal mapping $prox_{f \circ A}(x)$ can be evaluated efficiently by $$ (I + \partial (f \circ A))^{-1}(x) = prox_{f \circ A}(x) = A^T prox_{f}(A x), $$ as described here:

https://math.stackexchange.com/questions/147987/proximal-mapping-for-composition-of-functions

I was wondering if this extends to a proximal mapping with a diagonal matrix $D$ as step size: $$ (I + D \partial (f \circ A))^{-1}(x) = ? $$ I'm asking if the above proximal operator is easily to evaluate, assuming that $(I + D \partial f)^{-1}$ is easy.

I have been trying quite a while to find some closed-form for the above, but it seems difficult. I do not want to use an iterative method like ADMM to evaluate the proximal mapping.