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The convolution theorem states that the Fourier transform of the convolution of functions equals the pointwise multiplication of Fourier-transformed functions, i.e.:

$$\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$$

Are there other interesting $\mathcal{F}$ and $*$ for which the above holds, i.e. are there operations other than convolution that can be realized by pointwise multiplication, after some integral transform? I realize that 'interesting' is a bit vague, of course.

I understand that there are convolution theorems for Laplace and other similar transforms (under suitable conditions), but I'm unable find out about other operations and integral transforms.

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