Fourier inversion theorem

For the convention $\hat{f}(\xi )={\int }_{{\mathbb{R}}^n}{e}^{-ix\cdot \xi }f(x)dx$, if $f,\hat{f}\in L^1$, thenalmost everywhere. Reversing both exponential signs gives the equivalent opposite convention.

When $\hat{f}\in L^1$, its inverse Fourier integral is continuous by dominated convergence. If $f$ is also continuous and Fourier inversion identifies the two functions almost everywhere, then they agree everywhere.

For the angular-frequency convention and sufficient decay,

For the angular-frequency convention,Consequently cosine transforms to the half-sum of the two shifted deltas, while sine transforms to their signed difference divided by $2i$.

For the angular-frequency convention,so multiplication by a phase factor in frequency translates a function in physical space.