1 Introduction

We recall first a simple trigonometric identity:

$cos2t=-1+2{cos}^{2}t\text{ or equivalently}{cos}^{2}t=\frac{1}{2}+\frac{1}{2}cos2t$ (1)

Equation 1 can be interpreted as a simple finite Fourier series representation of the periodic function $f\left(t\right)={cos}^{2}t$ which has period $\pi$ . We note that the Fourier series representation contains a constant term and a period $\pi$ term.

A more complicated trigonometric identity is

${sin}^{4}t=\frac{3}{8}-\frac{1}{2}cos2t+\frac{1}{8}cos4t$ (2)

which again can be considered as a finite Fourier series representation. (Do not worry if you are unfamiliar with the result (2).) Note that the function $f\left(t\right)={sin}^{4}t$ (which has period $\pi$ ) is being written in terms of a constant function, a function of period $\pi$ or frequency $\frac{1}{\pi }$ (the “first harmonic”) and a function of period $\frac{\pi }{2}$ or frequency $\frac{2}{\pi }$ (the “second harmonic”).

The reason for the constant term in both (1) and (2) is that each of the functions ${cos}^{2}t$ and ${sin}^{4}t$ is non-negative and hence each must have a positive average value. Any sinusoid of the form $cosnt$ or $sinnt$ has, by symmetry, zero average value. Therefore, so would a Fourier series containing only such terms. A constant term can therefore be expected to arise in the Fourier series of a function which has a non-zero average value.