Advanced Calculus fi..

aalE506 Advanced Calculus, Fifth EditionIf we choose g(x) E 1 for XI 5 x 5 x2, we obtain the following result:COROLLARY Under the hypotheses of Theorem 15,4 that is, term-by-term integration is permissible for every Fourier series with respectto a complete orthogonal system {$,(x)).Although term-by-term integration causes no difficulties, term-by-term differentiationcalls for great caution. For example, the series sin x + . . . + (sin nxln) + . . .converges for all x, but the differentiated series cosx + . . . + cos nx + . . . divergesfor all x. Differentiation multiplies the nth term by n, which interferes withconvergence; integration divides the nth term by n, which aids convergence. Thesafest rule to follow is that of Theorem 33 of Section 6.14: Term-by-term differentiationof a convergent Fourier series is allowed, provided that the differentiatedseries converges uniformly in the interval considered.PROBLEMS1. Let&(x)=sinnx(n = 1,2, ...).a) Show that the functions (&(x)) form an orthogonal system in the interval 0 5 x 5 n.b) Show that the functions q5n(x) have the uniqueness property. [Hint: Let f (x) be afunction orthogonal to all the Let F(x) be the odd function coinciding with f (x)for 0 < x 5 n. Show that all Fourier coefficients of F(x) are 0.1c) Show that the Fourier sine series off (x) is uniformly convergent for every f (x) havingcontinuous first and second derivatives for 0 5 x 5 n and such that f (0) = f (n) = 0.d) Show that {&(x)} is a complete system for the interval 0 5 x 5 n.2. Cany out the steps (a), (b), (c), and (d) of Problem 1 for the functions &(x) = cosnx(n = 0, 1,2, ...). Show that the condition f (0) = f (n) = 0 IS not needed for (c).3. Prove the validity of (7.44) for functions f (x), g(x) that are piecewise continuous forasxsb.4. Verify the correctness of the Schwarz inequality (7.48) and the Minkowski inequality(7.49) for f (x) = x and g(x) = ex in the interval 0 5 x 5 1.5. Prove the corollary to Theorem 8 [see the proof of the corollary to Theorem 1, Section7.21.6. Prove Theorem 9 [cf. Problem 7 following Section 7.41.7. Prove the corollary to Theorem 9 [cf. the proof of the corollary to Theorem 2, Section 7.41.8. Prove Theorem 10. [Hint: Show by Parseval's equation (7.57) that if (h. 4,) = 0 for alln, then 11 h 11 = 0.19. a) Let the functions fn(x) be continuous for a 5 x 5 b and let the sequence fn(x) convergeuniformly to f (x) for a 5 x ( b. Prove that L.i.m.,,, fn(x) = f (x).b) Prove that the sequence cosn x converges to 0 in the mean for 0 5 x 5 n but doesnot converge for x = n. Prove that the sequence converges for 0 5 x 5 in but notuniformly.