A remark on the rigidity of a property characterizing the Fourier transform
Functional Analysis
2024-12-20 v1 Classical Analysis and ODEs
Abstract
We show rigidity results for the operator equations T(f.g) = Tf.Tg, T(f*g) = Tf.Tg and T(f.g) = Tf*Tg for bijective operators T acting on sufficently large spaces of smooth functions. Typically a condition like |T(f.g) - Tf.Tg| < a for all f, g with a fixed function a will imply T(f.g) = Tf.Tg. Theorems of Alesker, Artstein-Avidan, Faifman and Milman then yield characterizations (up to diffeomorphisms) of the Fourier transform by mapping products into convolutions and vice-versa on the Schwartz space.
Keywords
Cite
@article{arxiv.2412.14694,
title = {A remark on the rigidity of a property characterizing the Fourier transform},
author = {Hermann König and Vitali Milman},
journal= {arXiv preprint arXiv:2412.14694},
year = {2024}
}