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

$$ f (x) = sum limit _ { chi in hat {G}} c _ { chi} chi (x) tag {1} $$

where the $ c _ { chi} $ are complex numbers only determined satisfying $ sum limits | c _ { chi} | ^ 2 < infty $, and the right side converges to $ f $ in the $ L ^ 2 $-standard.

If more $$ sum limits | c _ { chi} | < infty tag {2} $$ then (1) is actually a point limit (and actually a uniform limit).

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

What is it when $ G = mathbb A_k / k $ for k $ a numeric field, and $ mathbb A_k $ the adeles of k $? There is a notion of smooth function on $ mathbb A_k $ (being smooth in the Archimedean argument and locally constant in the nonarchedian). Does the Fourier series of a smooth function $ f $ sure $ mathbb A_k / k $ to satisfy (2)? If not, is there a sufficient condition well known on $ f $ for (2) to hold?