论文标题
Collatz猜想和非架构的光谱理论 - 第二部分 - $ \ left（P，Q \右）$ - ADIC傅立叶分析和Wiener的Tauberian Theorem
The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- $\left(p,q\right)$-Adic Fourier Analysis and Wiener's Tauberian Theorem
论文作者
论文摘要
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.让$ k $是一个指标完整的，代数封闭的残留特征$ q $的本地领域，让$ c \ left（\ mathbb {z} _ {p}，k \ right）$是连续函数的banach空间$ \ mathbb {z} _ {z} _ {p} _ {p} _ {p} \ rightarrow k $ $ $\left(p,q\right)$-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 $dμ$的傅立叶 - 斯泰尔杰斯变换和$ \ mathbb {z} _ {p} $的所有点的radon-nikodym衍生物的非变化。
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 $K$.
