# Harmonic Analysis on the Ad\`ele Ring of $\Q$

## TL;DR

This paper develops a new harmonic analysis framework on the ring of finite adèles of the rationals, using ultrametric series representations, and extends it to the full adèle ring including the infinite prime.

## Contribution

It introduces a novel ultrametric series approach to adèle analysis and extends Fourier analysis techniques to the complete adèle ring of .

## Key findings

- Representation of adèles as series generalizing m-adic analysis
- New perspective on adelic Fourier analysis
- Extension of results to the complete adèle ring 

## Abstract

The ring of finite ad\`eles $\Af$ of the rational numbers $\Q$ is obtained in this article as a completion of $\Q$ with respect to a certain non--Archimedean metric. This ultrametric allows to represent any finite ad\`ele as a series generalizing $m$--adic analysis. The description made provides a new perspective for the adelic Fourier analysis on $\Af$, which also permits to introduce new oscillatory integrals. By incorporating the infinite prime, the mentioned results are extended to the complete ring of ad\`eles $\A$} of $\Q$.