# Absolute convergence of Fourier series in $\mathbb{Z}_p$

For a $$p$$-adic number $$a \in \frac{n}{p^k}+\mathbb{Z}_p\subset \mathbb{Q}_p$$ let $$\exp(2i \pi a) =\exp(2i \pi \frac{n}{p^k})$$ and $$\psi_a(x) = \exp(2i \pi ax)$$. Then $$Hom(\mathbb{Z}_p,\mathbb{C}^\times) = \{\ \psi_a , a \in \mathbb{Q}_p/\mathbb{Z}_p\}$$

• For $$f,h$$ (uniformly) continuous $$\mathbb{Z}_p \to \mathbb{C}$$ let $$\langle f,h \rangle =\lim_{k \to \infty} p^{-k} \sum_{n=0}^{p^k -1} f(n) \overline{h(n)}$$

• $$f \ast h(x) = \langle f(x-.),\overline{h} \rangle$$

• We obtain that $$\lim_{k \to \infty} \sum_{n=0}^{p^k -1} \langle f,\psi_{n/p^k}\rangle \psi_{n/p^k}(x) = \lim_{k \to \infty} f \ast \sum_{n=0}^{p^k -1} \psi_{n/p^k}(x)= \lim_{k \to \infty} f \ast p^k 1_{x \in p^k \mathbb{Z}_p} = f(x)$$ and the convergence is uniform.

• This is to be interpreted as a particular order of summation for the Fourier series $$\sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}\langle f,\psi_a\rangle \psi_a(x) \tag{1}$$

Question : when does $$(1)$$ converge absolutely ?

• If $$f$$ is locally constant, that is for some $$m$$, $$f(x) = f(x \bmod p^m)$$ then its Fourier series is a finite sum $$\sum_{a \in p^{-m}\mathbb{Z}_p/\mathbb{Z}_p}\langle f,\psi_a\rangle \psi_a(x)$$ as $$\langle f, \psi_{n/p^k} \rangle = 0$$ whenever $$k >m$$ ($$p\nmid n$$).

• With $$\|f\|_2^2 = \langle f,f \rangle$$ we have the Hilbert space $$L^2(\mathbb{Z}_p)$$ of which $$\{\ \psi_a , a \in \mathbb{Q}_p/\mathbb{Z}_p\}$$ is an orthonormal basis so $$f \in C^0(\mathbb{Z}_p) \implies f \in L^2(\mathbb{Z}_p) \implies \sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}|\langle f,\psi_a\rangle|^2 = \|f\|_2^2$$. Unlike the derivative $$L^2(\mathbb{R}/\mathbb{Z})\to L^2(\mathbb{R}/\mathbb{Z})$$ there is no obvious unbounded operator such that $$T(f) = \sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}c(a) \langle f,\psi_a\rangle \psi_a$$ and $$\sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}\frac{1}{|c(a)|^2} < \infty$$. If that operator could be easily defined we'd have $$T(f) \in L^2(\mathbb{Z}_p) \implies$$ $$\sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}|\langle f,\psi_a\rangle| \le (\sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}|\langle T(f),\psi_a\rangle|^2)^{1/2}(\sum_{a \in \mathbb{Q}_p/\mathbb{Z}_p}\frac{1}{|c(a)|^2})^{1/2} = C \|T(f)\|_2$$ so $$(1)$$ would converge absolutely.

$$\def\ZZ{\mathbb{Z}}\def\QQ{\mathbb{Q}}$$The Fourier series need not converge absolutely. First, some notation. Let $$A_n = p^n \ZZ_p \subseteq \ZZ_p$$ and let $$B_n = p^{-n} \ZZ_p/\ZZ_p \subset \QQ_p/\ZZ_p$$. Let $$\chi_S$$ denote the characteristic function of a set $$S$$, meaning that $$\chi_S(s)=1$$ if $$s \in S$$ and $$\chi_S(s)=0$$ otherwise. Then the Fourier transform of $$\chi_{A_n}$$ is $$p^{-n} \chi_{B_n}$$.
Let $$r_n$$ be any sequence of real numbers such that $$\sum_{n=0}^{\infty} r_n$$ is conditionally convergent. Then $$\sum_{n=0}^{\infty} r_n \chi_{A_n}$$ is a continuous function on $$\ZZ_p$$. (Because, if $$s \in A_m \setminus A_{m+1}$$, then $$\sum_{n=0}^{\infty} r_n \chi_{A_n}(s)=\sum_{n=0}^m r_n$$. If $$s \to 0$$ then $$m \to \infty$$, so this approaches $$\lim_{m \to \infty} \sum_{n=0}^m r_n$$, which is the definition of $$\sum_{n=0}^{\infty} r_n$$.)
The Fourier transform of $$\sum_{n=0}^{\infty} r_n \chi_{A_n}$$ is $$\sum_{n=0}^{\infty} r_n p^{-n} \chi_{B_n}$$. If $$t \in B_m \setminus B_{m-1}$$, then this is $$\sum_{n=m}^{\infty} r_n p^{-n}$$ (which is absolutely convergent, since $$r_n \to 0$$). If we sum over all points $$t \in B_m \setminus B_{m-1}$$, we get $$c_m:=(1-1/p) \sum_{n=m}^{\infty} r_n p^{m-n}.$$ If the Fourier series were absolutely convergent, then in particular we could clump together the $$t$$'s in the same $$B_m \setminus B_{m-1}$$ and then order the $$m$$'s arbitrarily. In other words, $$\sum c_m$$ would be aboslutely convergent.
But it is easy to come up with $$r_n$$ such that $$\sum c_m$$ is not absolutely convergent. For example, the standard $$\tfrac{(-1)^n}{n}$$ works.
• Tks a lot. So starting with $f(s) = \sum_{n\ge 0} \sum_{a \in \mathbb{Z}_p/p^n \mathbb{Z}_p} r(a,p^n) \chi_{a+p^n\mathbb{Z}_p}(s)$ and $\chi_{a+p^n\mathbb{Z}_p}(s)= \sum_{b \in B_n} p^{-n}\psi_b(-a)\psi_b(s)$ and $|r(a,p^n)| \le r_n$ then $\sum_{n \ge 0} r_n p^{n} < \infty$ implies the absolute convergence of the Fourier series $f(s) = \sum_{m \ge 0}\sum_{b \in B_m- B_{m-1} } \psi_b(s) ( \sum_{n\ge m}p^{-n} \sum_{a \in \mathbb{Z}_p/p^n \mathbb{Z}_p } r(a,p^n)\psi_b(-a))$. – reuns Jan 10 '19 at 5:45
• In particular $f : \mathbb{Z}_p \to \mathbb{C}$, $|f(s)-f(s_2)| \le C| s-s_2|_p^{1+\epsilon}$ implies the absolute convergence of the Fourier series. This seems like an acceptable starting point to manipulate Fourier series on $\widehat{\mathbb{Z}}$ and $\mathbb{A}/\mathbb{Q}$ (appearing in the Whittaker expansion of automorphic forms) – reuns Jan 10 '19 at 19:04