Advanced Calculus fi..

502 Advanced Calculus, Fifth Edition*7.12 SUFFICIENT CONDITIONS FOR COMPLETENESSTHEOREM 12 Let {$,(x)) be an orthogonal system of continuous functions forthe interval a p x 5 b. Let the following two properties hold: (a) {@,(x)} has theuniqueness property; (b) for some positive integer k, the Fourier series of g(x)with respect to {4,(x)} is uniformly convergent for every g(x) having continuousderivatives through the kth order for a 5 x 5 b and such that g(a) = gl(a) = ... =g(k'(a) = 0, g(b) = gl(b) = ... = g(k'(b) = 0. Then the system {4,,(x)} iscomplete.Proof. Let f (x) be piecewise continuous for a 5 x 5 b. We must then show that,given E > 0, a linear combination c, @l (x) + . . . + c,$, (x) = +(x) can be foundsuch that I ( f - $ 11 < E.For simplicity we assume the integer k of assumption (b) to be 2, as this case istypical.The construction of the function +(x) proceeds in several stages. We firstdetermine a continuous function F(x) such that 11 F - f 11< it and F(x) = 0 whena 5 x 5 a + 6, b - 6 p x 5 b for a proper choice of 6 > 0. This is suggested graphicallyin Fig. 7.14. We denote by fi(x) the piecewise continuous function that coincideswith f (x) except for a p x p a + 26 and for b - 26 p x p b, where fi(x)is identically zero. We now bridge the jumps in fl(x) by line segments. At eachjump the line joins [xo- 6, fi(xo - 6)] to [xo + 6, fl(xo + a)]. The function F(x)coincides with fi(x) except between xo - 6 and xo + 6, where it follows the straightline. Accordingly, F(x) is a continuous function, and 11 F - f 1l2 is a sum of a finitenumber K of integrals of formplus two integrals of [ f (x)I2 from a to a +6 and from b - 6 to b, where F = 0. Sincef (x) is piecewise continuous, If (x)( 5 M for some constant M. By its construction,I F(x)l 5 M also. Hence I F(x) - f (x)l 5 2M for every x, and@Figure 7.14 Approximation of the piecewise continuous function f (x)by a continuous function F(x) and by a smooth function G(x).