= Proximal operator under affine rescaling
{title2=$\operatorname{prox}_E(q)=\frac{y+\operatorname{prox}_{\alpha c^2J}(c(q-z)-y)}c$}
For a real <Hilbert space>, $E(x)=\alpha J(cx-y)+\langle x,z\rangle$, $\alpha>0$ and $c\ne0$, identify $z$ with its vector by the <Riesz representation theorem>. Completing the square in $\|x-q\|^2/2+\langle x,z\rangle$ shifts the input to $q-z$. Substitute $v=cx-y$ and multiply by $c^2$ to obtain the displayed formula, valid also for negative $c$. If $c=0$ and $J(-y)<\infty$, the answer is $q-z$; if $J(-y)=\infty$, the objective is identically infinite and no proper proximal problem remains.
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