A Fourier series is the expansion of any other function in terms of the Fourier functions of sine (nx) and cosine (nx). (These can also be expressed as a combination of positive and negative exponentials.) Since these functions compose a Hilbert space or a basis set, any continuous function can be composed of a combination of them as long as it has a finite number of finite discontinuities and a finite number of extrema in the interval over which the Fourier series is summed, usually 0 to 2 Pi. That is, as long as there's a set number of places where the function is not continuous, and as long as there are not an infinite number of local maxima and minima, the Fourier series can express the function. Each function will have a unique Fourier series expansion.

The Fourier coefficients, that is, the numbers multiplied by each sine...