Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms. Let ${\displaystyle \ f}$ and ${\displaystyle \ g}$ be two functions with convolution ${\displaystyle \ f*g}$. (Note that the asterisk denotes convolution in this context, and not multiplication. The tensor product symbol ${\displaystyle \otimes }$ is sometimes used instead.) Let ${\displaystyle \ {\mathcal {F}}}$ denote the Fourier transform operator, so ${\displaystyle \ {\mathcal {F}}\{f\}}$ and ${\displaystyle \ {\mathcal {F}}\{g\}}$ are the Fourier transforms of ${\displaystyle \ f}$ and ${\displaystyle \ g}$, respectively. Then

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