Fourier Series

32.521.510.5000.5 1 1.5 2 2.5 3xFigure 1.2.2(c). The three-term approximation.A comparison of the three figures and employment of the series identifies some importantpoints about the approximation procedure and it is sufficient to confine any remarks tox ≥ 0. Firstly, at x 0, the series gives S0 0, as sinn0 0 for all integer n, i.e. eachterm in the series is zero. Thus f0 S0 and the series converges (to zero) at this point.Thereafter, a better fit appears to be achieved (in some sense) with an increasing number ofterms in the trigonometric polynomials employed to approximate the series. Away fromx 0, the approximation takes the same value as fx on the same of occasions times as theorder of the polynomial.Finally, again from the series, it is clear that S 0sincesinn 0 for all integer n,whereas fx → as x → from the left-hand-side; so this is an issue that requires furtherinvestigation.16