Harmonic analysis on local fields and adelic spaces I

TL;DR

This paper develops harmonic analysis tools, including Fourier transform and Poisson formulas, for infinite-dimensional filtered vector spaces over finite fields, encompassing 2D local fields and adelic spaces of algebraic surfaces.

Contribution

It introduces a harmonic analysis framework on certain infinite-dimensional spaces over finite fields, extending classical analysis to new algebraic structures.

Findings

Fourier transform theory established for these spaces

Two-dimensional Poisson formulas derived

Framework applicable to algebraic surfaces over finite fields

Abstract

We develop a harmonic analysis on objects of some category $C_2$ of infinite-dimensional filtered vector spaces over a finite field. It includes two-dimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and two-dimensional Poisson formulas.