{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $\\omega$ be the space of all sequences in $\\mathbb{K}$.\nLet $A = (a_{nk})_{n, k \\ge 0}$ be an infinite matrix with entries in $\\mathbb{K}$, viewed as a [[matrix summability method|M6311DWJ1E]].\nFor $x = (x_k)_{k \\ge 0} \\in \\omega$, write $(Ax)_n = \\sum_{k \\ge 0} a_{nk} x_k$ when this series converges.\nThe *summability field* (or *convergence domain*) of $A$ is\n$$c_A = \\set{x \\in \\omega : (Ax)_n \\text{ converges for every } n, \\text{ and } \\lim_{n \\to \\infty} (Ax)_n \\text{ exists}}.$$\n\n## Remarks\n\nThe set $c_A$ is a [[sequence space|XYTN19EPF2]], since both conditions are preserved by sums and scalar multiples.\n\nExample: for the identity matrix, $(Ax)_n = x_n$, so $c_A$ is the space $c$ of [[convergent sequences|0W4CN9E2W5]].\n\nExample: for the Cesàro matrix, $a_{nk} = 1/(n+1)$ for $k \\le n$ and $a_{nk} = 0$ for $k \u003e n$, the sequence $x = (1, 0, 1, 0, \\dots)$ lies in $c_A$.\nIndeed $(Ax)_n$ is the proportion of ones among $x_0, \\dots, x_n$, which tends to $1/2$.\n\nNon-example: for the same matrix, $x_k = (-1)^k k$ is not in $c_A$.\nThe sum $x_0 + \\dots + x_n$ equals $n/2$ for even $n$ and $-(n+1)/2$ for odd $n$, so $(Ax)_n$ comes close to $1/2$ and to $-1/2$ infinitely often.\n\nA matrix $A$ is [[regular|29C7C41A52]] when $c \\subseteq c_A$ and $\\lim_n (Ax)_n = \\lim_k x_k$ for every $x \\in c$.\nWith a suitable topology, $c_A$ is an [[FK-space|71DMQHW6JZ]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "799DX4375M",
    "kind": "definition",
    "msc": [
      "40C05",
      "40H05"
    ],
    "name": "Summability field of a matrix method",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Silverman%E2%80%93Toeplitz_theorem"
      }
    ],
    "tag": "C22331MBY7",
    "uses": [
      "M6311DWJ1E",
      "XYTN19EPF2",
      "0W4CN9E2W5",
      "29C7C41A52",
      "71DMQHW6JZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/C22331MBY7/",
  "links": [
    {
      "field": "defines",
      "target": "799DX4375M"
    },
    {
      "field": "uses",
      "target": "M6311DWJ1E"
    },
    {
      "field": "uses",
      "target": "XYTN19EPF2"
    },
    {
      "field": "uses",
      "target": "0W4CN9E2W5"
    },
    {
      "field": "uses",
      "target": "29C7C41A52"
    },
    {
      "field": "uses",
      "target": "71DMQHW6JZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/C22331MBY7.md",
  "msc": [
    "40C05",
    "40H05"
  ],
  "name": "Summability field of a matrix method",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C22/C22331MBY7.md",
  "status": null,
  "tag": "C22331MBY7",
  "uri": "m2://C22331MBY7",
  "url": "https://math2.tamnd.com/wiki/summability-field-of-a-matrix-method-definition/",
  "writing": {
    "tag": "C22331MBY7",
    "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
      }
    ]
  }
}