A remark on the rigidity of a property characterizing the Fourier transform
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913618754273280 |
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| author | König, Hermann Milman, Vitali |
| author_facet | König, Hermann Milman, Vitali |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A remark on the rigidity of a property characterizing the Fourier transform König, Hermann Milman, Vitali Functional Analysis Classical Analysis and ODEs 39B42, 26B05, 47A62 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. |
| title | A remark on the rigidity of a property characterizing the Fourier transform |
| topic | Functional Analysis Classical Analysis and ODEs 39B42, 26B05, 47A62 |
| url | https://arxiv.org/abs/2412.14694 |