= Inverse Fourier transform
{title2=$\mathcal F^{-1}g(x)=\frac1{2\pi}\int_{\mathbb R}g(k)e^{ikx}\,dk$}
With the <Fourier transform> convention $\widehat f(k)=\int_{\mathbb R}f(x)e^{-ikx}\,dx$, the inverse Fourier transform is the displayed <linear operator>. The integral is defined directly for an <integrable function> $g$. The <Fourier inversion theorem> ensures $\mathcal F^{-1}\widehat f=f$ under its hypotheses, in particular for <Schwartz functions>, and for integrable $f$ and $\widehat f$ at continuity points of $f$. The operation also extends to <tempered distributions>; the normalization and sign depend on the chosen <Fourier transform> convention.
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