Abstract
This paper gives an overview of \(\left(p,q\right)\) -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 \(\left(p,q\right)\) -adic generalization of Norbert Wiener’s celebrated Tauberian Theorem. Letting \(K\) be a metrically complete, algebraically closed local field of residue characteristic \(q\) , letting \(C\left(\mathbb{Z}_{p},K\right)\) be the Banach space of continuous functions \(\mathbb{Z}_{p}\rightarrow K\) , and letting \(d\mu\) be a \(\left(p,q\right)\) -adic measure (a continuous linear functional \(C\left(\mathbb{Z}_{p},K\right)\rightarrow K\) ), the \(\left(p,q\right)\) -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 \(K\) .
