The Fourier series
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?
Re: The Fourier series
Can you be more specific as to what you don't understand? It's talking about how a Fourier Series exists for any function over a given interval. For example consider a square wave function - you can find the series that is equivalent to a repeating square wave from t=0 to infinity, including the interval from t=0 to 10 seconds. Now imagine a second function consisting of a square wave from t=0 to t=10, then at t=10 seconds it changes to a sawtooth - you can find the Fourier Series that runs from t=0 to t=20 consisting of the square wave and then the sawtooth wave, and it's a different series than the first one, as that was a pure square wave. So now we have two different series that both result in a square wave for the first ten seconds. With this reasoning you can see that there are an infinite number of functions that have a square wave shape from t=0 to t=10 but have some other shape after t=10. Hence there are an infinite number of Fourier Series that consist of a square wave from t = 0 to t=10. Does this help?