Filtering signals equals multiplying their spectra by the filter function.

Evidence

A powerful theorem behind the Fourier transform is the convolution theorem. Convolution is what filters do when they filter signals. However, we normally think about the action of the filter as multiplication of the spectrum of the signal by the filtering function. For example, a low-pass filter gets rid of the high frequencies and passes low frequencies. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 2769

Categorization

• XREF: Ties directly to the book’s earlier filter chapters — convolution in time is multiplication in frequency.