{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $k \\ge 1$, let $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_k = b$, and let $y_0, \\dots, y_k \\in \\R$.\nLet $s$ be the natural [[cubic spline|V42H0R3EKX]] with knots $x_0, \\dots, x_k$ such that $s(x_i) = y_i$ for all $i$.\nLet $g \\colon [a, b] \\to \\R$ be twice continuously differentiable with $g(x_i) = y_i$ for all $i$.\nThen\n$$\\int_a^b s''(x)^2 \\, dx \\le \\int_a^b g''(x)^2 \\, dx,$$\nwith equality only if $g = s$.\n\n## Remarks\n\nLet $h = g - s$, which vanishes at every knot.\n[[Integration by parts|08Y1BBQTK9]] gives $\\int_a^b s'' h'' \\, dx = \\bigl[s'' h'\\bigr]_a^b - \\int_a^b s''' h' \\, dx$.\nThe boundary term is $0$ because $s''(a) = s''(b) = 0$.\nOn each $[x_{i-1}, x_i]$ the function $s'''$ is a constant $c_i$, so the last integral is $\\sum_i c_i \\bigl(h(x_i) - h(x_{i-1})\\bigr) = 0$.\nHence $\\int g''^2 = \\int s''^2 + \\int h''^2$, and equality forces $h'' = 0$, so $h$ is linear with two zeros and $h = 0$.\n\nExample: for two data points ($k = 1$) the natural cubic spline is the straight line through them.\nThe statement says that every other $C^2$ curve through the two points has $\\int g''^2 \u003e 0$.\n\nThe end conditions matter.\nFor other cubic spline interpolants, such as the clamped one, the inequality holds only among functions $g$ that satisfy the same end conditions.\n\n## Explanations\n\n### L1 human\n\nThe integral of the squared second derivative measures how much a curve bends overall; for a thin elastic strip it is roughly its bending energy.\nThe theorem says that among all smooth curves through given points, the natural cubic spline bends the least.\nThis is why it imitates the draftsman's spline, a flexible strip held by pegs at the data points.\n\n### L2 human\n\nThe theorem gives a variational characterisation of the natural cubic spline interpolant.\nIt minimises $\\int_a^b g''^2$ among the $C^2$ functions $g$ that satisfy the interpolation conditions.\nThe same holds in the larger Sobolev space $H^2$.\nThe proof shows an orthogonality: $s''$ is orthogonal to $h''$ for every $h$ vanishing at the knots, and this orthogonality is what makes $s$ the minimiser.\nReplacing the interpolation conditions by a penalised fit leads to [[smoothing splines|BNEXZF75QC]], and the same idea in two variables leads to [[thin plate splines|P9ZPH7EC8K]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "V42H0R3EKX"
    ],
    "kind": "statement",
    "msc": [
      "41A15",
      "65D07",
      "41A05"
    ],
    "name": "Minimal curvature property of natural cubic splines",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spline_interpolation"
      }
    ],
    "tag": "28C5WWGX9J",
    "uses": [
      "V42H0R3EKX",
      "D59HBKKRP9",
      "0W29TBTY12",
      "08Y1BBQTK9"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/28C5WWGX9J/",
  "links": [
    {
      "field": "about",
      "target": "V42H0R3EKX"
    },
    {
      "field": "uses",
      "target": "V42H0R3EKX"
    },
    {
      "field": "uses",
      "target": "D59HBKKRP9"
    },
    {
      "field": "uses",
      "target": "0W29TBTY12"
    },
    {
      "field": "uses",
      "target": "08Y1BBQTK9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/28C5WWGX9J.md",
  "msc": [
    "41A15",
    "65D07",
    "41A05"
  ],
  "name": "Minimal curvature property of natural cubic splines",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/28C/28C5WWGX9J.md",
  "status": null,
  "tag": "28C5WWGX9J",
  "uri": "m2://28C5WWGX9J",
  "url": "https://tamnd.github.io/math2/wiki/minimal-curvature-property-of-natural-cubic-splines/",
  "writing": {
    "tag": "28C5WWGX9J",
    "kind": "statement",
    "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
      }
    ]
  }
}