I understand how to calculate the fourier coefficients and I understand the importance of orthogonality of sines and cosines. But why can any periodic functio be expressed as a linear combination of sines and cosines? Fourier series is such a common and important subject in math, but I can not find a clear explanation why it is possible?
The Taylor expansion somehow makes sense, as you use derivatives to calculate the function. But I don't see how the fourier series work. Could someone explain this clearly in not too opaque theorems?