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  "body": "\n## Definition\n\nLet $G$ be an additive abelian group, let $Y$ be a set, and let $h \\in G$.\nThe *shift operator* $E_h$ with step $h$ sends a [[function|0XHDTE5YMZ]] $f \\colon G \\to Y$ to the function\n$$E_h f \\colon G \\to Y, \\qquad E_h f(x) = f(x + h).$$\nFor a [[sequence|0QE8STGJH2]] $u = (u_n)_{n \\ge 0}$ in $Y$, the (left) shift $E$ is defined by $(E u)_n = u_{n+1}$ for $n \\ge 0$.\n\n## Remarks\n\nExample: for $f(x) = \\sin x$ and $h = \\pi/2$, $E_h f(x) = \\sin(x + \\pi/2) = \\cos x$.\nNon-example: $f \\mapsto f(2x)$ is not a shift operator; it rescales the argument instead of translating it.\nThe shifts compose by $E_h E_k = E_{h+k}$, and $E_0$ is the identity.\nWhen $Y$ is an abelian group, $E_h$ is additive, and the [[forward difference operator|8E0EBA8X07]] is $\\Delta_h = E_h - I$.\nOther notations are $T_h$ and $\\tau_h$; on sequences the left shift is also called the backward shift.\n",
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