{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $1 \\le p \u003c \\infty$.\nFor a function $g \\colon \\R \\to \\C$ whose $p$-th power is locally integrable, put\n$$\\norm{g}_{B^p} = \\Big( \\limsup_{T \\to \\infty} \\frac{1}{2T} \\int_{-T}^{T} \\abs{g(x)}^p \\, dx \\Big)^{1/p} \\in [0, \\infty].$$\nA *generalized trigonometric polynomial* is a function $P(x) = \\sum_{k=1}^{N} c_k e^{i\\lambda_k x}$ with $N \\ge 1$, $\\lambda_k \\in \\R$ and $c_k \\in \\C$.\nLet $f \\colon \\R \\to \\C$ be a function with $\\abs{f}^p$ locally integrable.\nIt is *Besicovitch almost periodic* of order $p$, or $B^p$ almost periodic, if the following holds.\nFor every $\\varepsilon \u003e 0$ there is a generalized trigonometric polynomial $P$ with $\\norm{f - P}_{B^p} \u003c \\varepsilon$.\n\n## Remarks\n\nThe quantity $\\norm{\\cdot}_{B^p}$ is a seminorm, not a norm.\nIt vanishes on every function in the [[Lebesgue space|SASP14MA1Z]] $L^p(\\R)$, and the space of $B^p$ classes is obtained by identifying functions $f, g$ with $\\norm{f - g}_{B^p} = 0$.\nEvery [[Bohr almost periodic function|ED69H4C3PK]] is $B^p$ almost periodic for every $p$, since $\\norm{g}_{B^p} \\le \\sup \\abs{g}$.\nFor $p = 2$, each $B^2$ almost periodic $f$ has a mean value of $\\abs{f}^2$ and Fourier coefficients $a(\\lambda)$, nonzero for at most countably many $\\lambda$, with $\\lim_{T \\to \\infty} \\frac{1}{2T} \\int_{-T}^{T} \\abs{f}^2 = \\sum_\\lambda \\abs{a(\\lambda)}^2$.\n\nExample: $f = 1_{[0,1]}$ is $B^p$ almost periodic for every $p$.\nIndeed $\\frac{1}{2T} \\int_{-T}^{T} \\abs{f}^p = \\frac{1}{2T}$ for $T \\ge 1$, so $\\norm{f - 0}_{B^p} = 0$ and the polynomial $P = 0$ works.\nThis $f$ is not continuous, so it is not Bohr almost periodic.\n\nNon-example: $f(x) = x$ is not $B^p$ almost periodic.\nEvery generalized trigonometric polynomial $P$ is bounded, say by $C$, so $\\abs{f - P} \\ge \\abs{x} - C$ and $\\norm{f - P}_{B^p} = \\infty$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "BA9HBM8DFR",
    "kind": "definition",
    "msc": [
      "42A75",
      "43A60"
    ],
    "name": "Besicovitch almost periodic function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Almost_periodic_function"
      }
    ],
    "tag": "JR7PBS4WAR",
    "uses": [
      "SASP14MA1Z"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/JR7PBS4WAR/",
  "links": [
    {
      "field": "defines",
      "target": "BA9HBM8DFR"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/JR7PBS4WAR.md",
  "msc": [
    "42A75",
    "43A60"
  ],
  "name": "Besicovitch almost periodic function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JR7/JR7PBS4WAR.md",
  "status": null,
  "tag": "JR7PBS4WAR",
  "uri": "m2://JR7PBS4WAR",
  "url": "https://tamnd.github.io/math2/wiki/besicovitch-almost-periodic-function-definition/",
  "writing": {
    "tag": "JR7PBS4WAR",
    "kind": "definition",
    "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
      }
    ]
  }
}