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

We show rigidity results for the operator equations T(f.g) = Tf.Tg, T(fg) = Tf.Tg and T(f.g) = TfTg 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.