{
  "id": "2412.14694",
  "version": "v1",
  "published": "2024-12-19T09:53:14.000Z",
  "updated": "2024-12-19T09:53:14.000Z",
  "title": "A remark on the rigidity of a property characterizing the Fourier transform",
  "authors": [
    "Hermann König",
    "Vitali Milman"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-19T09:53:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B42",
      "26B05",
      "47A62"
    ],
    "keywords": [
      "fourier transform",
      "property characterizing",
      "schwartz space",
      "operator equations",
      "yield characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}