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  "body": "\n## Definition\n\nLet $G$ and $H$ be additive abelian groups, and let $h \\in G$.\nThe *backward difference operator* with step $h$ sends a function $f \\colon G \\to H$ to the function $\\nabla_h f \\colon G \\to H$ given by\n$$\\nabla_h f(x) = f(x) - f(x - h) \\qquad (x \\in G).$$\nWhen $h = 1$, one writes $\\nabla$ for $\\nabla_1$.\nFor a sequence $u \\colon \\Z \\to H$, this gives $(\\nabla u)_n = u_n - u_{n-1}$.\n\n## Remarks\n\nIt is a [[finite difference|7JSQZ126BP]] in the backward direction.\nExample: for $f(x) = x^2$ on $\\R$, $\\nabla_h f(x) = x^2 - (x - h)^2 = 2hx - h^2$.\nNon-example: $f(x + h) - f(x)$ is in general not $\\nabla_h f(x)$; for $f(x) = x^2$ it equals $2hx + h^2$.\nThe two operators are related by $\\nabla_h f(x) = \\Delta_h f(x - h)$, where $\\Delta_h$ is the [[forward difference operator|8E0EBA8X07]].\nIn operator form, $\\nabla_h = I - E_{-h}$, with $E_{-h} f(x) = f(x - h)$ a [[shift operator|MBK6AE6734]].\n",
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