The Fourier series can be used to represent an arbitrary function within the interval from - π to + π even though function does not continue or repeat outside this interval. Outside this interval the Fourier series expression will repeat faithfully from period to period irrespective of whether the given function continues. The same remarks apply to any arbitrary function which is specified over any finite range, say from t=0 to t=t0. An infinite number of Fourier series expansion with fundamental periods T≥t0 can be found such that they all reproduce f(t) within the given range. Outside this range, different expansion may have entirely different values, depending upon the choice of T as compared with t0 and of the waveform in the interval from t= t0 to T, which is entirely arbitrary except that the Dirichlet conditions must be satisfied. I was reading a book, Analysis of linear systems-by David K. Cheng, to learn Fourier series. I have understood every think except the above writing. Could someone please explain me the purpose of the above writing?