Fourier analysis was the decomposition of a periodic signal in terms of a sum of sinusoidal basis functions (versus their frequencies).
Data noted that he knew that his mother was an android because, as he stated, "the intervals between blinks were governed by the Fourier system, the same mathematical formula my father used to give my blinking pattern the appearance of randomness." (TNG: "Inheritance")
In 2378, The Doctor played The Blue Danube waltz in one of his recitals. When he and Captain Janeway were abducted by aliens known as the Hierarchy, The Doctor altered a recording of the waltz so that it would sound out of tune. By performing a Fourier analysis, Seven of Nine found out that the harmonics of the piece had been modified to imitate a warp signature. The crew was able to track the signature and thereby locate The Doctor and the captain. (VOY: "Renaissance Man")