Fourier analysis is the decomposition of a function in terms of a sum of sinusoidal basis functions (versus their frequencies) that can be recombined to obtain the original function. That process of recombining the sinusoidal basis functions is also called Fourier synthesis (in which case Fourier analysis refers specifically to the decomposition process).
In 1996, Rain Robinson performed a Fourier analysis at the Griffith Observatory, when Tom Paris told her that her "curves don't look so great", in relation to the analysis. (VOY: "Future's End")
Data's blinking was governed by the Fourier analysis, to simulate 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")