{
  "id": "1807.01847",
  "version": "v1",
  "published": "2018-07-05T04:51:38.000Z",
  "updated": "2018-07-05T04:51:38.000Z",
  "title": "Symmetric Decompositions of $f\\in L^2(\\mathbb{R})$ Via Fractional Riemann-Liouville Operators",
  "authors": [
    "Yulong Li"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "It is proved that given $-1/2<s<1/2$, for any $f\\in L^2(\\mathbb{R})$, there is a unique $u\\in \\widehat{H}^{|s|}(\\mathbb{R})$ such that $$ f=\\boldsymbol{D}^{-s}u+\\boldsymbol{D}^{s*}u\\,, $$ where $\\boldsymbol{D}^{-s}, \\boldsymbol{D}^{s*}$ are fractional Riemann-Liouville operators and the fractional derivatives are understood in the weak sense. Furthermore, the regularity of $u$ is discussed, and other versions of the results are established. As an interesting consequence, the Fourier transform of elements of $L^2(\\mathbb{R})$ is characterized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-05T04:51:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A33",
      "34A08",
      "46N20"
    ],
    "keywords": [
      "fractional riemann-liouville operators",
      "symmetric decompositions",
      "fractional derivatives",
      "weak sense",
      "fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}