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](https://mathoverflow.net/questions/319911/question-about-the-fourier-expansion-of-adelic-eisenstein-series-for-operatorn).



Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$
Observe that, because we can cover by open sets, on each of which the function is constant under translation by some open in the finite adeles, by compactness we can take a finite subcover, do the function is translation-invariant by the intersection, another open subgroup. Modding our the whole adele group by this, and quotienting by the field, we just get avtorus.


So the problem reduces to the Torus case, which is well-known.


