Much to my dismay, in my work the more number-theoretic side of harmonic analysis (ex: the fourier transform on the adeles, on the profinite integers, etc.), I have found myself struggling with technicalities that emerge from frustratingly simple issues—so simple (and yet, so technical) that I haven’t been able to find anything that might shine any light on the matter.

Let $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ be the space of functions $ f:\mathbb{Q}\rightarrow\mathbb{C}$ which satisfy $ \sum_{t\in T}\left|f\left(t\right)\right|^{2}<\infty$ for all bounded subsets $ T\subseteq\mathbb{Q}$ . Letting $ \mu$ be any positive integer, it is easy to show that any functions which is both $ \mu$ -periodic (an $ f:\mathbb{Q}\rightarrow\mathbb{C}$ such that $ f\left(t+\mu\right)=f\left(t\right)$ for all $ t\in\mathbb{Q})$ and in $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ is necessarily an element of $ L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ , the complex hilbert space of functions $ f:\mathbb{Q}/\mu\mathbb{Z}\rightarrow\mathbb{C}$ so that: $ $ \sum_{t\in\mathbb{Q}/\mu\mathbb{Z}}\left|f\left(t\right)\right|^{2}<\infty$ $ Equipping $ \mathbb{Q}/\mu\mathbb{Z}$ with the discrete topology, we can utilize Pontryagin duality to obtain a Fourier transform: $ \mathscr{F}_{\mathbb{Q}/\mu\mathbb{Z}}$ . The ideal case is when $ \mu=1$ . There, $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}$ is an isometric hilbert space isomorphism from $ L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)$ to $ L^{2}\left(\overline{\mathbb{Z}}\right)$ , where: $ $ \overline{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$ $ is the ring of profinite integers, where $ \mathbb{P}$ is the set of prime numbers, and where $ L^{2}\left(\overline{\mathbb{Z}}\right)$ is the space of functions $ \check{f}:\overline{\mathbb{Z}}\rightarrow\mathbb{C}$ which are square-integrable with respect to the haar probability measure $ d\mathfrak{z}=\prod_{p\in\mathbb{P}}d\mathfrak{z}_{p}$ on $ \overline{\mathbb{Z}}$ .

The first sign of trouble was when I learned that, for any integers $ \mu,\nu$ , the (additive) quotient groups $ \mathbb{Q}/\mu\mathbb{Z}$ and $ \mathbb{Q}/\nu\mathbb{Z}$ are group-isomorphic to one another, and thus, that both have $ \overline{\mathbb{Z}}$ as their Pontryagin dual. From my point of view, however, this isomorphism seems to only cause trouble. In my work, I am identifying $ L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ with the set of $ \mu$ -periodic functions $ f:\mathbb{Q}\rightarrow\mathbb{C}$ which are square integrable with respect to the counting measure on $ \mathbb{Q}\cap\left[0,\mu\right)$ . As such, $ \mathbb{Q}/\mu\mathbb{Z}$ and $ \mathbb{Q}/\nu\mathbb{Z} $ cannot be “the same” from my point of view, because $ f\left(t\right)\in L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ need not imply that $ f\left(t\right)\in L^{2}\left(\mathbb{Q}/\nu\mathbb{Z}\right)$ .

In my current work, I am dealing with a functional equation of the form:$ $ \sum_{n=0}^{N-1}g_{n}\left(t\right)f\left(\frac{a_{n}t+b_{n}}{d_{n}}\right)=0$ $ where $ N$ is an integer $ \geq2$ , where the $ g_{n}$ s are known periodic functions, where $ f$ is an unknown function in $ L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)$ (i.e., $ f\left(t\right)\in L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ and $ f\left(t+1\right)=f\left(t\right)$ for all $ t\in\mathbb{Q})$ , and where $ a_{n},b_{n},d_{n}$ are integers with $ \gcd\left(a_{n},d_{n}\right)=1$ for all $ n$ . For brevity, I’ll write: $ $ \varphi_{n}\left(t\right)\overset{\textrm{def}}{=}\frac{a_{n}t+b_{n}}{d_{n}}$ $ Because $ \varphi_{n}\left(t+1\right)$ need not equal $ \varphi_{n}\left(t\right)+1$ , the individual functions $ f\circ\varphi_{n}$ , though periodic and in $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ , are not necessarily going to be of period $ 1$ . Letting $ p$ denote the least common multiple of the periods of $ g_{n}$ and the $ f\circ\varphi_{n}$ s, I can view the functional equation as existing in $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ , and as such, I hope to be able to simplify it by applying the fourier transform.

Letting $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ denote the duality pairing between elements $ t\in\mathbb{Q}$ (or $ \mathbb{Q}/\mu\mathbb{Z}$ ) and $ \mathfrak{z}\in\overline{\mathbb{Z}}$ , the idea is to multiply the functional equation by $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ , sum over an appropriate domain of $ t$ (ideally, $ \mathbb{Q}/p\mathbb{Z}$ ), make a change of variables in $ t$ to move one of the $ \varphi_{n}\left(t\right)$ s out of $ f$ and into $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ , pull out terms from this character, and then invert the fourier transform to return to $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ with a vastly simpler equation. My main difficulty can be broken into three parts:

(1) Is taking the least common multiple of the periods to reformulate the functional equation as one over $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ legal?

(2) I know that the fourier transform $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}:L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)\rightarrow L^{2}\left(\overline{\mathbb{Z}}\right)$ is given by:$ $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{t\in\mathbb{Q}/\mathbb{Z}}f\left(t\right)e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ $ and that the inverse transform is: $ $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}^{-1}\left\{ \check{f}\right\} \left(t\right)\overset{\textrm{def}}{=}\int_{\overline{\mathbb{Z}}}\check{f}\left(\mathfrak{z}\right)e^{-2\pi i\left\langle t,\mathfrak{z}\right\rangle }d\mathfrak{z}$ $ However, I am at a loss as to what formula to use for $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}$ and its inverse, and for two reasons. On the one hand, because $ \mathbb{Q}/\mathbb{Z}$ and $ \mathbb{Q}/p\mathbb{Z}$ are group-isomorphic, what is to stop me from using the same formula for their fourier transforms? On the other hand, if I use a modified formula—say:$ $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{t\in\mathbb{Q}/p\mathbb{Z}}f\left(t\right)e^{2\pi i\left\langle \frac{t}{p},\mathfrak{z}\right\rangle }$ $ does the fact that $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)\in L^{2}\left(\overline{\mathbb{Z}}\right)$ then mean that I can recover $ f$ by applying $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}^{-1}$ , or do I have to also modify it in order to make everything consistent? Knowing the correct formula for the fourier transform on $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ and its inverse is essential.

(3) I would like to think that performing a change-of-variables for a sum of the form:$ $ \sum_{\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)$ $ (where $ \alpha,\beta,r\in\mathbb{Q}$ , with $ \alpha\neq0$ and $ r=\frac{p}{q}>0$ ) would be a relatively simple matter, but, that doesn’t appear to be the case. For example, if $ t\in\mathbb{Q}\rightarrow f\left(\alpha t+\beta\right)\in\mathbb{C}$ is not a $ r$ -periodic function, then this sum is not well-defined over the quotient group $ \mathbb{Q}/r\mathbb{Z}$ . Worse yet—supposing $ f$ is $ r$ -periodic take a look at this: write elements of $ \mathbb{Q}/r\mathbb{Z}$ in co-set form: $ t+r\mathbb{Z}$ , where $ t\in\mathbb{Q}$ . Then, make the change-of-variable $ \tau=\alpha t+\beta$ . Consequently, the set of all $ \tau$ is:$ $ \alpha\left(\mathbb{Q}/r\mathbb{Z}\right)+\beta=\left\{ \alpha\left(t+r\mathbb{Z}\right)+\beta:t\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:t\in\mathbb{Q}\right\}$ $ . Here is where things get loopy.

(1) Since $ \alpha,\beta\in\mathbb{Q}$ with $ \alpha\neq0$ , the map $ \varphi\left(t\right)\overset{\textrm{def}}{=}\alpha t+\beta$ is a bijection of $ \mathbb{Q}$ . As such, I would think that:$ $ \left\{ \tau+\alpha r\mathbb{Z}:t\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:\varphi^{-1}\left(\tau\right)\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:\tau\in\varphi\left(\mathbb{Q}\right)\right\}$ $ and hence:$ $ \alpha\left(\mathbb{Q}/r\mathbb{Z}\right)+\beta=\left\{ \tau+\alpha r\mathbb{Z}:\tau\in\mathbb{Q}\right\} =\mathbb{Q}/\alpha r\mathbb{Z}$ $ Using this approach, I obtain:$ $ \sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=\sum_{\tau\in\mathbb{Q}/\alpha r\mathbb{Z}}f\left(\tau\right)$ $

(2) Since $ r=\frac{p}{q}$ , decompose $ \mathbb{Z}$ into its equivalence classes mod $ q$ :$ $ \left\{ \tau+\alpha r\mathbb{Z}:\tau\in\mathbb{Q}\right\} =\bigcup_{k=0}^{q-1}\left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\}$ $ and so:$ $ \sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=\sum_{k=0}^{q-1}\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau+\alpha rk\right)$ $ On the other hand, for each $ k$ :$ $ \left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\} =\left\{ \tau+\alpha rk+\alpha rq\mathbb{Z}:\tau\in\mathbb{Q}\right\}$ $ and, since $ \tau\mapsto\tau+\alpha rk$ is a bijection of $ \mathbb{Q}$ , the logic of (1) would suggest that:$ $ \left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\} =\left\{ \tau+\alpha rq\mathbb{Z}:\tau\in\mathbb{Q}\right\}$ $ for all $ k$ . But then, that gives:$ $ \sum_{k=0}^{q-1}\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau+\alpha rk\right)=\sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=q\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau\right)$ $ which hardly seems right.