The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $\left(p,q\right)$-Adic Fourier Analysis and Wiener's Tauberian Theorem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909654682959872 |
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| author | Siegel, Maxwell C. |
| author_facet | Siegel, Maxwell C. |
| contents | 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 $dμ$ 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 $dμ$'s Fourier-Stieltjes Transform and the non-vanishing of the Radon-Nikodym derivative of $dμ$ at all points in $\mathbb{Z}_{p}$ where the derivative exists in $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11082 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $\left(p,q\right)$-Adic Fourier Analysis and Wiener's Tauberian Theorem Siegel, Maxwell C. General Mathematics 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 $dμ$ 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 $dμ$'s Fourier-Stieltjes Transform and the non-vanishing of the Radon-Nikodym derivative of $dμ$ at all points in $\mathbb{Z}_{p}$ where the derivative exists in $K$. |
| title | The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $\left(p,q\right)$-Adic Fourier Analysis and Wiener's Tauberian Theorem |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2208.11082 |