Math equation for finding frequency-domain component coefficients in a time-domain signal

Title says it all: we're taking an input signal (such as a sound wave), downshifting it in the frequency domain by the frequency of interest, and mean-filtering that to find the frequency-domain component coefficient. Particularly, we are interested in finding the piano notes that are being played in some input audio, but this is just the math derivation: we'll do the actual software prototyping for this in a separate video.