Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $L$ with respect to $\langle\;,\; \rangle$. We let $L^*$ denote the dual lattice of $L$ with respect to $\langle\;,\rangle$. Let $T:=\mathbf{R}^n/L$ be the corresponding real torus endowed with the Lebesgue measure dx. For every integrable function $f:T\rightarrow \mathbf{C}$ and each $\xi\in L^*$ we let $$ a_{\xi}(f):=\frac{1}{cov(L)}\int_T f(x)e^{-2\pi i\langle x,\xi\rangle}dx $$ denote the $\xi$ Fourier coefficient of $f$. Assume now that $f$ is a real analytic function on $T$. Does this imply that there exists positive constants $c,C\in\mathbf{R}_{>0}$ such that $$ |a_{\xi}(f)|\leq Ce^{-c||\xi||}\;\; $$ for all $\xi\in L^*$ ?

Here $||\xi||$ corresponds to the norm of the vector $\xi\in L^*$. If such a result is true I would like to have a reference. I seem to remember that this is true when $n=1$. I would also be interested to know if one can go in the other direction namely if $f$ is a smooth function (infinitely many times differentiable) such that its Fourier coefficients satisfy an inequality as above then $f$ is real analytic.