Convolution in one domain...

Z-transforms taught us that a multiplication of polynomials is a convolution of their coefficients. Interpreting $Z=e^{i\omega}$ as having numerical values for real numerical values of $\omega$, we have the idea that

Convolution in the time domain is multiplication in the frequency domain.

Expressing Fourier transformation as a matrix, we see that except for a choice of sign, Fourier transform is essentially the same process as as inverse fourier transform. This creates an underlying symmetry between the time domain and the frequency domain. We therefore deduce the symmetrical principle that

Convolution in the frequency domain is multiplication in the time domain.