Harmonic Analysis on the Ad\`ele Ring of
Abstract
The ring of finite ad\`eles of the rational numbers is obtained in this article as a completion of 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 , 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 of .
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