Fourier Series

hx Afx Bgx ,for constants A and B, will also have the period L 0 . This may seem to be a trivial result butit is not; it shows how an infinite number of periodic functions may be constructed fromjust two periodic functions. Thus for any period L 0 , there will be an infinite number offunctions with this period and the properties of a two-term expression will be carried overto a longer sum or one of infinite length.10.5-1-0.5000.51x-0.5-1y =f(x)y = g(x)Figure 1.1.2(a). The functions fx x 2 and gx x 3 .We can also approach periodic functions from another viewpoint. Suppose that thefunction fx is only defined on −1,1 and that it is well-behaved on this interval. Notethat for the moment, this is restricted to the open interval −1,1 rather than the closedinterval −1,1; the reason for this will become apparent shortly. As an example considerthe two functions fx x 2 and gx x 3 , shown in figure 1.1.2(a), and which were givenas examples of even and odd functions earlier.The two functions can be extended to −, by defining two new functions to coincidewith fx and gx on −1,1 and to be periodic with period 2, shown in figures 1.1.2(b) and5