{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "A7Q6581B1F"
    },
    {
      "field": "uses",
      "target": "Z72JWTXM3W"
    },
    {
      "field": "defines",
      "target": "FAMZ5Z5JK4"
    }
  ],
  "body": "\n## Summary\n\nA Besov space $B^s_{p,q}(\\R^n)$ is a space of functions or [[tempered distributions|AES2T5JPFP]] that measures smoothness of order $s$ in the $L^p$ sense.\nA third index $q$ gives a fine tuning.\nIt is defined by cutting the [[Fourier transform|DPVM34ADZV]] of a function into dyadic frequency shells and measuring each piece in $L^p$.\nBesov spaces contain many classical spaces, such as Hölder spaces and fractional Sobolev spaces, and they are stable under real [[interpolation|VX1R6FN6PS]].\nThey appear naturally as spaces of boundary values of Sobolev functions and in the study of nonlinear evolution equations.\n\n## Explanations\n\n### L1 human\n\nA function can be split into pieces that oscillate at frequencies of size about $1, 2, 4, 8, \\ldots$.\nSmoothness means that the high-frequency pieces are small.\nThe Besov norm asks that the piece at frequency $2^j$ be smaller than $2^{-js}$ in the $L^p$ sense, in a summable way controlled by $q$.\nFor $p = q = 2$ this is the familiar condition $\\int (1 + \\abs{\\xi}^2)^s \\abs{\\hat f(\\xi)}^2 \\, d\\xi \u003c \\infty$ on the Fourier transform.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "46E35",
      "42B35"
    ],
    "name": "Besov space",
    "same_as": {
      "wikidata": "Q526261",
      "wikipedia": "Besov space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Besov_space"
      }
    ],
    "tag": "95QBBPRJ9E"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/95QBBPRJ9E/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/95QBBPRJ9E.md",
  "msc": [
    "46E35",
    "42B35"
  ],
  "name": "Besov space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/95Q/95QBBPRJ9E.md",
  "status": null,
  "tag": "95QBBPRJ9E",
  "uri": "m2://95QBBPRJ9E",
  "url": "https://tamnd.github.io/math2/wiki/besov-space/",
  "writing": {
    "tag": "95QBBPRJ9E",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}