Can you tell me a biological problem which will be solved only by using Fourier series? Please bring the problem here and mention to its solve.
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Oscillation phenomena are abundant in biology (e.g. cell cycle, circadian rhythms) and Fourier analysis is very helpful, or sometimes requisite, for analyzing them.
The following is a recent paper in PNAS that explores this concept in the system of P53-Mdm2 oscillation in human cells. (P53 is a very famous tumor suppressor gene, and Mdm2 is another protein that degrades P53.)