The Fourier transform on valuations is the Fourier transform

Dmitry Faifman, Thomas Wannerer*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number110741
JournalJournal of Functional Analysis
Volume288
Issue number3
DOIs
StatePublished - 1 Feb 2025

Funding

FundersFunder number
Département de Mathématiques et de Statistique, Université de Montréal
Israel Science Foundation1750/20
Deutsche ForschungsgemeinschaftWA 3510/3-1

    Keywords

    • Convex bodies
    • Fourier transform
    • Homogeneous distributions
    • Valuations

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