Comparison theorem between Fourier transform and Fourier transform with compact support
type: Item
instance of: scholarly article
title: Comparison theorem between Fourier transform and Fourier transform with compact support
zbMATH Open document ID: 1285.14022
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2013-06-11T00:00:00Z
review text: This paper is of interest to those studying number theory. It compares the Fourier transform and the Fourier transform with compact support in the context of arithmetic \(\mathcal{D}\)-modules. Recall that for a function \(f: \mathbb{R} \rightarrow \mathbb{C}\) the classical Fourier transform is defined by \[ \hat{f}(\omega) = \int_{\mathbb{R}}f(t)e^{-\omega t} dt \] where \(\omega\) is on the imaginary axis. The Fourier transform satisfies the identities \(\widehat{f'}=w\hat{f}\) and \((\hat{f})'=\widehat{-tf}\). Considering multiplication by \(\omega\) and \(t\) as endomorphisms of the sheaf of smooth functions, the Fourier transform interchanges \(t \leftrightarrow -\frac{d}{d\omega}\) and \(\frac{d}{dt} \leftrightarrow \omega\). Identifying the local coordinates \(t\) and \(\omega\) on the real and imaginary axis as a single variable \(x\) and generalizing to dimension \(n\), the Fourier transform induces an algebraic automorphism of the Weyl algebra \(\mathcal{D}_{\mathbb{R}^n}=\mathbb{R}\langle x_1,\dots,x_n, \partial_1,\dots,\partial_n\rangle\) by \(x_i \mapsto -\partial_i\) and \(\partial_i \mapsto x_i\). This can be thought of as a noncommutative change of coordinates on the cotangent space \(T^*\mathbb{R}^n\). The Fourier transform of a \(\mathcal{D}\)-module \(\mathcal{M}\), \(\widehat{\mathcal{M}}\), is defined as the pushforward along this endomorphism. This is a categorification of the process of solving differential equations by replacing differential operators with polynomials. The Fourier transform functor can be reformulated in terms of pullback functor and a pushforward functors on a certain diagram. This permits definitions of the Fourier transform in a variety of algebraic, analytic, and arithmetic settings. It also helps to show that variations such as Fourier transform for constructible sheaves and Fourier transform with compact supports are reasonably defined. In the setting of constructible \(\ell\)-adic sheaves, \textit{N. M. Katz} and \tex…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 43A25
zbMATH DE Number: 6173999
zbMATH Keywords: arithmetic \(D\)-modules
zbMATH Keywords: Fourier Transform
author: Christine Noot-Huyghe
MaRDI profile type: Publication
full work available at URL: jtnb.827
OpenAlex ID: W2325592251
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DOI: 10.5802/JTNB.827
description: scientific article; zbMATH DE number 6173999