Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion

$$f(x) = \sum\limits_{\chi \in \hat{G}} c_{\chi} \chi(x) \tag{1}$$

where the $c_{\chi}$ are uniquely determined complex numbers satisfying $\sum\limits |c_{\chi}|^2 < \infty$, and the right hand side converges to $f$ in the $L^2$-norm.

If moreover $$\sum\limits |c_{\chi}| < \infty \tag{2}$$ then (1) is actually a pointwise limit (and in fact a uniform limit).

When $G = \mathbb R/\mathbb Z$, it is well known that a sufficient condition for (2) is that $f$ be smooth (even just $C^1$).

What about when $G = \mathbb A_k/k$ for $k$ a number field, and $\mathbb A_k$ the adeles of $k$? There is a notion of a smooth function on $\mathbb A_k$ (being smooth in the archimedean argument, and locally constant in the nonarchimedean). Does the Fourier series of a smooth function $f$ on $\mathbb A_k/k$ satisfy (2)? Or if not, is there a well known sufficient condition on $f$ for (2) to hold?

The example I have in mind is the Fourier expansion of Eisenstein series, which I've asked about in a previous question here.