Translation of a function ($T_{h}f(x)=f(x-h)$)

A translation of a function shifts its function input without altering its values relative to that shift. On the real line, $(T_{h}f)(x)=f(x-h)$ moves the graph of a function rightward by $h$: if $f$ has its peak at zero, $T_{h}f$ has it at $h$. These shifts satisfy $T_h{T}_k=T_{h+k}$ and $T_{-h}=T_h^{-1}$.

Translation invariance of Lebesgue measure preserves each finite Lp norm. For the Fourier transform convention $\hat{f}(\xi )={\int }_{\mathbb{R}}f(x)e^{-ix\xi }dx$, a change of variables givesThe 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.