A remark on the rigidity of a property characterizing the Fourier transform

TL;DR

This paper investigates the rigidity of certain operator equations related to the Fourier transform, showing that approximate conditions imply exact characterizations of the Fourier transform through operator equations on smooth functions.

Contribution

It establishes that near-identity operator equations imply the operator is essentially the Fourier transform, providing new rigidity results and characterizations.

Findings

Operator equations characterize the Fourier transform up to diffeomorphisms.

Approximate operator relations imply exact Fourier transform properties.

Rigidity results extend to operators on large spaces of smooth functions.

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.