Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 493so that f (x) is represented by a trigonometric series for all x. The series is the Fourierseries of f (x), for1= -3T = S" -7rbn = - 1 f (x) sin nx dx = - [g(x) + s F(x)] sin nx dxand similarly, a, = A,. Therefore the Fourier series of f (x) converges to f (x) forall x. At x = 0 the series converges to f (0), which was defined to be the averageof left and right limits at x = 0. Since the series for g(x) is uniformly convergentfor all x, while the series for F(x) converges uniformly in each closed interval notcontaining x = 0 (or x = 2kn), the Fourier series of f (x) converges uniformly ineach such closed interval.The theorem has now been proved for the case of just one jump discontinuity. Ifthere are several jumps, at points xl, x2, . . . , we simply remove them by subtractingfrom f (x) the functionThe resulting function g(x) is again continuous and piecewise very smooth, so thatthe same conclusion holds. Thus the theorem is established in all generality.Remarks. The proof just given uses the principle of superposition: The Fourierseries of a linear combination of two functions is the same linear combination ofthe corresponding two series. This can be put to very good use to systematize thecomputation of Fourier series. Illustrations are given in Problem 1 which follows.The idea of subtracting off the series corresponding to a jump discontinuityalso has a practical significance. If a function f (x) is defined by its Fourier seriesand is not otherwise explicitly known, one can, of course, use the series to tabulatethe function. If f (x) has a jump discontinuity, the convergence will be poor nearthe discontinuity; this will reveal itself in the presence of terms having coefficientsapproaching O like lln. If the discontinuity xl and jump sl are known, as is oftenthe case, one can subtract the corresponding function sl F(x -XI) as before; the newseries will converge much more rapidly.The same idea can be applied to functions f (x) that are continuous but for whichfl(x) has a jump discontinuity sl at xl. One now subtracts from f the functionslG(x - xl), for this continuous function has as derivative precisely the functionsl F(x - xl), with jump sl at xl . By integrating G(x) - 21713, one obtains a periodiccontinuous function having a jump in its second derivative at x = 0. Continuing inthis way, one provides a jump function for each derivative; each such function canbe used to remove corresponding slowly converging terms from the Fourier series.PROBLEMS1. Let fi(x) and f2(x) be defined by the equations:fi(x) = 0, - x 0 fi(x) = 1, Oixln;f2(~) =0, -n~x