# Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$

Let $$p$$ and $$q$$ be prime numbers. When $$p=q$$, Mahler's Theorem gives a complete description of $$C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$$, the space of continuous functions from $$\mathbb{Z}_{p}$$ to $$\mathbb{Z}_{p}$$. I'm wondering (possibly in vain) if there might be a comparable classification of $$C\left(\mathbb{Z}_{p};\mathbb{Z}_{q}\right)$$ when $$p$$ and $$q$$ are distinct.

I ask only because I've been doing $$p$$-adic harmonic analysis, but have found myself having to brave the wilds of $$L^{\infty}\left(\mathbb{Z}_{p};\mathbb{C}_{q}\right)$$, the space of all $$f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$$ so that:$$\sup_{\mathfrak{z}\in\mathbb{Z}_{p}}\left|f\left(\mathfrak{z}\right)\right|_{q}<\infty$$

Pontryagin duality lets me do Fourier analysis on $$L^{\infty}\left(\mathbb{Z}_{p};\mathbb{C}\right)$$; for $$p=q$$, on the other hand, I can use things like the volkenborn integral, or the amice transform / mazur-mellin transform—$$p$$-adic distributions, in general. The problem is, without a structure theorem like Mahler's for the $$p\neq q$$ case, though I can define “integration” on $$L^{\infty}\left(\mathbb{Z}_{p};\mathbb{C}_{q}\right)$$ by elements of its dual space (continuous functionals $$\varphi:L^{\infty}\left(\mathbb{Z}_{p};\mathbb{C}_{q}\right)\rightarrow\mathbb{C}_{q})$$, I don't see a way to do useful computations for the specific, non-abstract functions that I'm trying to fourier analyze.

So, I guess what I'm really asking is: how do you take the "integral" or "fourier transform" of such a function?

Any thoughts? Reference recommendations? Etc.?

$${\mathbb Z}_p$$ and $${\mathbb Z}_q$$ are homeomorphic; hence so are $$C({\mathbb Z}_p,{\mathbb Z}_p)$$ and $$C({\mathbb Z}_p,{\mathbb Z}_q)$$.