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.
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.)