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arXiv · 2412.14694

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

Hermann König·Vitali Milman
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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.

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Hermann König, Vitali Milman. 2024-12-19. A remark on the rigidity of a property characterizing the Fourier transform. https://arxiv.org/abs/2412.14694

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