= Translation of a function
{title2=$T_hf(x)=f(x-h)$}
= Function translation
{synonym}
= Translations of a function
{synonym}
A <translation of a function> shifts its <function input> without altering its values relative to that shift. On the <real line>, $(T_hf)(x)=f(x-h)$ moves the <graph of a function> rightward by $h$: if $f$ has its peak at zero, $T_hf$ has it at $h$. These shifts satisfy $T_hT_k=T_{h+k}$ and $T_{-h}=T_h^{-1}$.
<Translation invariance> of <Lebesgue measure> preserves each finite <Lp norm>. For the <Fourier transform> convention $\widehat f(\xi)=\int_{\mathbb R}f(x)e^{-ix\xi}\,dx$, a <change of variables> gives
$$
\widehat{T_hf}(\xi)=e^{-ih\xi}\widehat f(\xi).
$$
The phase sign depends on the stated shift and <Fourier transform> conventions. In particular, a spatial shift changes phase while preserving the <modulus> of the transform.
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