In 1D, the Fourier series $u(x)=\sum_{k\in\mathbb{Z}}\hat{u}_ke^{ikx}$ is analytic on the torus if and only if there exist constants $K>0$ and $a>0$ such that $$|\hat{u}_k|\leq K e^{-a|k|}.$$ This is the subject of for example this post. Does this generalize to multiple Fourier series on the form $u(x)=\sum_{k\in \mathbb{Z}^2} \hat{u}_k e^{ik\cdot x}$ for $x\in \mathbb{T}^2$ and a condition $$|\hat{u}_k|\leq Ke^{-a\|k\|}?$$
Is there a reference, or how do I show this?
I am not entirely convinced by the argument in 1D: Here it is pointed out that the corresponding Laurent series converges in an annulus containing the unit circle (which we identify with the torus) in the complex plane. But why does make the Fourier series analytic on $\mathbb{T}$? Is that just a matter of identification?
Here’s another 1D argument which may convince you a bit better.
Suppose that $u(x)=\sum_{k}{\hat{u_k}e^{ikx}}$ is analytic on the torus. Then there is some $r>0$ and a $2\pi$-periodic analytic function $f(z)$ defined on $|\Im{z}| < r$ such that $f(x)=u(x)$ when $x \in \mathbb{R}$.
Let $n >0$, then $\hat{u_n}=\int_0^1{u(2\pi x)e^{-2i\pi nx}\,dx}$. Fix some $r > s>0$: then it’s easy to see that $\hat{u_n}=\int_{-si}^{1-si}{f(2\pi z)e^{-2i\pi z}\,dz}$ (that’s because the contour integral of $f(2\pi z)e^{-2i\pi n z}$ on the rectangle with vertices $0,1,-si,1-si$ vanishes).
But for $z \in [-si,1-si]$, $|e^{-2i\pi n z}| \leq e^{-2\pi n s}$, so that $|\hat{u_n}| \leq C(e^{-2\pi s})^{n}$.
A similar argument (with the path $si -> 1+si$) gives another exponentially-decreasing bound for $|\hat{u_n}|$ when $n <0$.
Now assume that we have a function $u(x \in \mathbb{T}^2)=\sum_{k \in \mathbb{Z}^2}{\hat{u_k}e^{i k\cdot x}}$ which is real-analytic. Then there is some $r$ and a holomorphic function $f(z)$, defined for $|\max(\Im{z_1},\Im{z_2})| < r$, such that $f$ is $\mathbb{Z}^2$-periodic, and $f(x \in \mathbb{R}^2)=u(x)$.
Assume that $k_1,k_2>0$ and fix $r>s>0$. Then, as above, $\hat{u_k}=\int_{-si,1-si}{\int_{-si}^{1-si}{u(z_1,z_2)e^{-2i\pi k_1z_1}e^{-2i\pi k_2z_2}\,dz_2}\,dz_1}$. The same estimate as above shows that $|\hat{u_k}| \leq Ce^{-2\pi s(k_1+k_2)}$. We can do the exact same thing for other signs with different paths, as in the 1D case, yielding a general bound $\hat{u_k}=O(e^{-a|k|})$ for some $a>0$.