{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "Y7JP4NA725"
    }
  ],
  "body": "\n## Summary\n\nA de Branges space is a Hilbert space of [[entire functions|0419794R0T]] built from one entire function $E$.\nThe function $E$ must be larger at each point of the upper half-plane than at the mirror point in the lower half-plane.\nIts elements are the entire functions $F$ such that $F/E$ and $F^\\#/E$ lie in the Hardy space of the [[upper half-plane|X6XGGH5FPW]].\nThe norm of $F$ is the $L^2$ norm of $F/E$ on the real line.\nIt is a [[reproducing kernel Hilbert space|4CK2GSRAGD]] with an explicit kernel, and it generalises the Paley–Wiener spaces.\nDe Branges used these spaces in the spectral theory of canonical systems.\n\n## Explanations\n\n### L1 human\n\nChoose an entire function $E$ with $\\abs{E(\\bar z)} \u003c \\abs{E(z)}$ whenever $z$ is in the upper half-plane.\nThe space $\\mathcal{H}(E)$ collects entire functions that are \"no larger than $E$\" in a precise square-integrable sense.\nFor $E(z) = e^{-iaz}$ with $a \u003e 0$, one gets the Paley–Wiener space of square-integrable band-limited functions, whose Fourier transforms live on $[-a, a]$.\nEach point evaluation $F \\mapsto F(w)$ is a bounded functional, given by an inner product with an explicit kernel.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46E22",
      "30D20"
    ],
    "name": "de Branges space",
    "same_as": {
      "wikidata": "Q5244279",
      "wikipedia": "De Branges space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/De_Branges_space"
      }
    ],
    "tag": "NYHNTP9AND"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/NYHNTP9AND/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/NYHNTP9AND.md",
  "msc": [
    "46E22",
    "30D20"
  ],
  "name": "de Branges space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NYH/NYHNTP9AND.md",
  "status": null,
  "tag": "NYHNTP9AND",
  "uri": "m2://NYHNTP9AND",
  "url": "https://math2.tamnd.com/wiki/de-branges-space/",
  "writing": {
    "tag": "NYHNTP9AND",
    "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
      }
    ]
  }
}