The Fourier transform is an integral transform with turns any function into a superposition of sinusoidal waves. The convolution theorem states the astonishing property that if you convolve two functions, this is equivalent to simply multiplying the corresponding transformed functions. Since the inverse Fourier transform is nearly identical to the Fourier transform, it is perhaps unsurprising that the same holds in the other direction.
Do all integral transforms have this fascinating property? Or is it unique to the Fourier transform? (And if the Fourier transform is the only integral transform in all existence that has this property, why is that?)