# Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$

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.

• Okay, if I understand correctly, you pull back $f$ to a function on $\mathbb A$, and claim using compactness that there is an open compact subgroup $H$ of $\mathbb A_f$ such that $f(x+h) = f(x)$ for all $x \in \mathbb A$ and $h \in H$. Then, $f$ becomes well defined on $\mathbb A/H = \mathbb A_{\infty} \times \mathbb A_f/H$. Is this correct? – D_S Jan 22 '19 at 4:33
• I think I made a mistake when going from $\mathbb{Z}_p$ to $\widehat{\mathbb{Z}}$. The criterion I obtain for the absolute convergence of the Fourier series of $f : \widehat{\mathbb{Z}} \to \mathbb{C}$ should be $|f(x)-f(y)| \le C N(x-y)^{1+\epsilon}$ where $N(x) = | \widehat{\mathbb{Z}}/x \widehat{\mathbb{Z}} |= \prod_p |x_p|_p^{-1}$ @D_S – reuns Jan 22 '19 at 15:02
• Obtained from $f(x) = \sum_n \sum_{a \in \widehat{\mathbb{Z}}/n! \widehat{\mathbb{Z}}}c(a,n)\chi_{a+n! \widehat{\mathbb{Z}}}(x)$ and $\chi_{a+n! \widehat{\mathbb{Z}}}(x) = \sum_{s \in (n!)^{-1}\widehat{\mathbb{Z}}/\widehat{\mathbb{Z}}} \psi_s(-a)\psi_s(x)$ and $\psi$ the character $\mathbb{A_Q}_{fin} /\widehat{\mathbb{Z}} \to \mathbb{C}^\times$, $\psi(p^{-k}) = \exp(2i \pi p^{-k})$ – reuns Jan 22 '19 at 15:07