Continuity on the torus

Let $\mathbb{T}^d$ be the $d$-dimensional torus, for all $r >0,\eta_r(x):=e^{-r|x|^2},x \in \mathbb{T}^d,$ we denote by $\mathscr{F}^{-1}\eta_r$ the inverse Fourier transform defined by $\mathscr{F}^{-1}\eta_r(x):=\sum_{k \in \mathbb{Z}^d}\eta_r(k)e^{2\pi\mathrm{i}\langle x,k\rangle},x \in \mathbb{T}^d.$ Let $v_0 \in C^{\infty}(\mathbb{T}^d),$

Is it true that $$\forall x \in \mathbb{T}^d,\lim_{r \to 0}\mathscr{F}^{-1}\eta_r*v_0(x)=v_0(x)?$$

I know how to prove this when we are dealing with $\mathbb{R}^d:$$$\forall x \in \mathbb{R}^d,\int_{\mathbb{R}^d}v_0(x-y)\mathscr{F}^{-1}\eta_r(y)dy=\int_{\mathbb{R}^d}\mathscr{F}^{-1}\eta_1(y)v_0(x-ry)dy$$ where we used a simple linear change of variable, we conclude using the dominated convergence theorem, that the limit is $v_0(x).$

How to treat the problem when we are on the torus? Unfortunately, the above change of variable won't work.