The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $(p,q)$-Adic Fourier Analysis and Wiener's Tauberian Theorem

This paper gives an overview of $(p,q)$-adic Fourier theory - the Fourier theory of functions from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes - which we then use to prove a novel $(p,q)$-adic generalization of Norbert Wiener’s celebrated Tauberian Theorem. Letting $K$ be a metrically complete, algebraically closed local field of residue characteristic $q$, letting $C({\mathbb{Z}}_p,K)$ be the Banach space of continuous functions ${\mathbb{Z}}_p\to K$, and letting $d\mu$ be a $(p,q)$-adic measure (a continuous linear functional $C({\mathbb{Z}}_p,K)\to K)$, the $(p,q)$-adic Wiener Tauberian Theorem (WTT) we prove establishes the equivalence of the density of the span of translates of $d\mu$’s Fourier-Stieltjes Transform and the non-vanishing of the Radon-Nikodym derivative of $d\mu$ at all points in ${\mathbb{Z}}_p$ where the derivative exists in $K$.