Inverse Fourier transform (${\mathcal{F}}^{-1}g(x)=\frac{1}{2\pi }{\int }_{\mathbb{R}}g(k)e^{ikx}dk$)

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