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Norbert Wiener's celebrated Tauberian Theorem has many equivalentformulations; among these, it gives necessary and sufficient conditions for thespan of translates of functions to be dense in $L^{1}$. This paper establishestwo generalizations of Wiener's Tauberian Theorem to the setting of$\left(p,q\right)$-adic analysis, the study of functions$\chi:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{q}$, where $\mathbb{Z}_{p}$ and$\mathbb{C}_{q}$ are the $p$-adic integers and $q$-adic complex numbers,respectively, for distinct primes $p$ and $q$. The first generalization regardscontinuous $\left(p,q\right)$-adic functions; the second regards$\left(p,q\right)$-adic measures, equivalently, continuous $q$-adic-valuedlinear functionals on the space $C\left(\mathbb{Z}_{p},\mathbb{C}_{q}\right)$of all continuous $\left(p,q\right)$-adic functions. Together, these two$\left(p,q\right)$-adic Wiener Tauberian Theorems provide a way of studying thevalues attained by $\left(p,q\right)$-adic functions using$\left(p,q\right)$-adic Fourier analysis.

Wiener Tauberian Theorems and the Value Distributions of $\left(p,q\right)$-adic Functions