The Fourier transform on valuations is the Fourier transform

TL;DR

This paper provides a direct description of Alesker's Fourier transform on valuations using the classical Fourier transform on functions, simplifying proofs and confirming a key conjecture about its properties.

Contribution

It introduces a direct description of Alesker's Fourier transform on valuations via the classical Fourier transform, confirming a previously conjectured property.

Findings

Simplified proofs of main properties of the Alesker--Fourier transform.

First proof of a conjectured property of the Alesker--Fourier transform.

Established a direct link between valuations' Fourier transform and classical Fourier transform.

Abstract

Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant smooth valuations that has the same functorial properties as the classical Fourier transform. In this paper, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker--Fourier transform. One of these properties was previously only conjectured by Alesker and is proved here for the first time.