Ark9: Exercises for MAT2400 Fourier series The exercises o this sheet cover the sectios 4.9 to 4.13. They are iteded for the groups o Thursday, April 12 ad Friday, March 30 ad April 13. NB: No group o Thursday, March 29! The distributio is the followig: Friday, March 30 ad April 13: No 1, 2, 3, 4, 7, 9, 10, 12, 13, 14, 15. The rest for Thursday, April 12. Key words: Periodic fuctios, Fourier series, Dii s test, Itegratio of Fourier series. Periodic fuctios, the spaces D ad C P Problem 1. ( Tom s otes 4.8, Problem 1 (page 122)). Show that C P is a closed subset of C([ π, π], C). Problem 2. ( Tom s otes 4.8, Problem 2 (page 122)). I this problem we shall prove some properties of the space D of piecewise cotiuous fuctios o [ π, π]. if f,g are fuctios from D, the f + g D ad fg D. b) Show that D is a vector space. c) Show that all fuctios i D are bouded. d) Show that all fuctios i D are Riema itegrable o [ π, π]. e) Show that f,g = 1 2 Problem 3. π π f(x)g(x) dx is a ier product o D. a) Let the fuctio f o [0, 2π] be give by f(x) =x π. Let f be the periodic extesio of f to R with period 2π. Describe what f looks like, whe restricted to [ π, π]. b) If g is defied i [π, 3π] byg(x) =x π ad g deotes the 2π-periodic extesio of g to R, what does the restrictio of g to [ π, π] look like? Problem 4. Let f(x) be the fuctio defied o the etire real lie R which is periodic with period 2π ad which is equal to x whe x ( π, π) ad has f(π) =f( π) =0. x 2 = (+1) si x,
Ark9: Fourier series MAT2400 sprig 2012 wheever x ( π, π). What happes i the edpoits? Show further that f(x) 2 = (+1) si x for ay x R. Hit: Use Dii s test ad the periodicity. b) Let g be the 2π-periodic fuctio g(x) = f(x π). Show that x + π if 0 <x π g(x) = f(x π) = x π if π x<0 0 if x =0 c) Use problem 5 o Ark8 to verify that we have g(x) 2 = si x for x ( π, π). d) Show that x 2 = π 2 si x wheever x (0, 2π). Explai why this is ot i cotradictio with??. Riema-Lebesgue lemma ad Dii s test Problem 5. Let I =[a, b] [ π, π] be a iterval. Let χ I be the characteristic fuctio of I, i.e., the fuctio such that χ I (x) = 1 if x I ad χ I (x) = 0 if x I. the Fourier coefficiets of χ I is give by ad show that lim ± c =0. b) Show that c = i 2π (e ib e ia ), χ I (x) =(a b)/2π + 1 π 1 (si b si a)cosx +(cosa cos b) si x) 2
Ark9: Fourier series MAT2400 sprig 2012 if x = a, b. What happes if x = a or x = b? Recall that a step fuctio o [ π, π] is a fuctio s(x) which is a fiite liear combiatio of characteristic fuctios of itervals cotaied i [ π, π], i.e., s(x) = 1 k r a kχ Ik (x) where the I k s are itervals cotaied i [ π, π] fork =1,...,r. c) If we deote by c the -th Fourier coefficiet of a step fuctio s, show that lim ± c =0. Problem 6. Let f(x) be a bouded ad Riema itegrable fuctio o [ π, π]. The almost by defiitio of the Riema itegral if >0 is give, we ca fid a step fuctio s(x) such that π π f(x) s(x) dx <. Use this to show that lim ± c (f) = 0, where c (f) is the -th Fourier coefficiet of f. Hit:?? ca be usefull. Problem 7. Let f(x) be a periodic fuctio with period 2π which is p times cotiuously differetiable for all x R, ad let c be its -th Fourier coefficiet. Show that lim p c =0. Hit: Use partial itegratio ad iductio o p. Problem 8. Let f(x) be a bouded, Riema itegrable fuctio o [ π, π], ad let g(x) = x f(t) dt. Show that the two oesided limits 0 g(x+ ) ad g(x ) exist for ay x ( π, π), ad that the Fourier series of g coverges to (g(x + )+g(x ))/2 for all x [ π, π]. Hit: Use Diis test. Problem 9. a) Let f(x) be itegrabel over the iterval [ π, π]. Assume that there is a ope iterval I [ π, π] such that f(x) = 0 for all x I. Show that the Fourier series of f coverges to 0 at all x I. b) Let ow f(x) ad g(x) be two itegrabel fuctios o [ π, π] ad assume that there is a ope iterval I [ π, π] such that f(x) =g(x) forx I. Show that if y I the the Fourier series of f coverges at y if ad oly if the Fourier series of g coverges there. I case the series coverge, show that their sums are equal. Problem 10. a) Let f(x) be a fuctio o [ π, π] belogig to the space D. Show that if all the Fourier coefficiets of f are zero, the f is idetical zero. Hit: Parseval s idetity may be usefull. 3
Ark9: Fourier series MAT2400 sprig 2012 b) Show that if two fuctios from D have the same Fourier coefficiets, they are equal. Problem 11. Let h (0,π). h = si h cos x =0, wheever x ( π, π) ad π> x >h. What happes i the poits x = ±π ad x = ±h? Check the formulas you obtai by givig x these values agaist earlier formulas (i.e., from problem?? o this Ark). Hit: Problem 10 o Ark8. b) Explai why the equatio?? at the first view seems to be a paradox. The explai why??, after a secod though, i fact is ot a paradox. Hit: Use the formula 2 si α cos β = si(α + β) + si(α β) ad the Fourier series from problem??. ( ) Itegratio of Fourier series Problem 12. a) Fid the mea value of f(x) = x2 4 b) Show that we have the equality over the iterval [ π, π]. for x [ π, π]. Hit: x 2 = c) Show that π2 12 = Problem 13. for all x R. b) Show that for all x R. x 2 4 = π2 12 + cos x 2 +1. 2 +1 si x (x 3 π 2 x)/12 = i [ π, π]. si x (x 4 2π 2 x 2 )/48 = 7π4 720 + +1 cos x 4 4 3
Ark9: Fourier series MAT2400 sprig 2012 c) Determie +1 1 4. Problem 14. Asume that a trigoometric series = c e ix coverges uiformly. the series is its ow Fourier series. b) Why is?? ot a utterly stupid questio? Problem 15. for x ( π, π). si x 2 = 8 π +1 si x 4 2 1 b) Itegrate the series?? to obtai the Fourier series for cos x 2. c) Differetiate the series?? to obtai the Fourier series for cos x 2. d) Check that the two series thus obtaied are equal. ( ) Example of a trigoometric series that is ot a Fourier series Problem 16. the trigoometric series =2 si x log ( ) coverges for all x. Hit: Use Dirichlet s criterio, problem 1 o Ark7. b) Compute the series oe obtais by termwise itegratio of the series??. c) Show that the itegrated series diverges for x =0. d) Show that the series i?? is ot the Fourier series of ay fuctio i D. Hit: Use propositio 14.13.1 i Tom s otes. e) Let f(x) be the fuctio defied by??, i.e., f(x) = si x =2. Show that f(x) is log cotiuous if x = 0.Hit: Use Dirichlet s criterio to see that?? coverges uiformly o itervals of the form I =(δ, π) ad ( π, δ), where δ>0. The behavior of si x ad cos x whe Problem 17. The aim of this exercise is two prove the followig, which is about ratioal approximatio of irratioal umbers eeded to uderstad how si x ad cos x behaves for big. 5
Ark9: Fourier series MAT2400 sprig 2012 Let a irratioal umber y ad N a atural umber be give. The there is a atural umber N ad a iteger k such that y k < 1 N+1. a) For ay iteger r betwee 1 ad N let f(r) =ry ry. 1. Show that 0 <f(r) < 1 for ay of the r s, ad that f(r) = f(r ) if r = r. Hece the set A = {f(r) :1 r N} {0, 1} is cotaied i [0, 1] ad has N + 2 elemets. Hit: Show that ay of the two cases f(r) =f(r )orf(r) = 0 implies that y is ratioal. b) Divide I = [0, 1] ito the N + 1 itervals I s =[ s 1, s )fors =1,...,N ad N+1 N+1 I N+1 =[ N 1, 1]. Show that for some s betwee 1 ad N +1 at least two of the members N+1 of A are cotaied i I s. Coclude by showig that (r r )y (ry r y) < 1 N+1. The techique used i the last part, is sometimes called the pigeohole priciple, i.e., if you are to place a certai umber of pigeos i a certai umber of pigeoholes ad you have more pigeos tha holes, the at least two pigeos must be placed i the same hole. Problem 18. This problem deals with the result that if x is ot a ratioal multiple of 2π, the B x = {x +2kπ : Z,k Z} is dese i R. if z B x the z B x. b) Use problem?? to show that for ay >0 there is a member z B x such that z <. Hit: Take a look at x/2π. c) Let B x deote the closure of B x, show that R = B x. Hit: Let U be the complemet of B x.ifu U ad u>0, let α = if{a : There is a b such that u (a, b) U}. Check that α>0, ad get a cotradictio by??. Problem 19. Let x be a umber that is ot a ratioal multiple of 2π. the two sets {si x : N} ad {cos x : N} are dese i [ 1, 1]. b) Show that either of the limits lim si x ad lim cos x exists. Show that the series cos x ad si x both diverge. c) Let =0 b si x be a trigoometric series, ad assume that all the coefficiets b are positive. Show that the series has ifiitly may positive ad ifiitely may egative terms. Hit: Use??. Versjo: Moday, March 26, 2012 12:08:28 PM 1 If z is ay real umber, z deotes the greatest iteger less tha or equal to z; e.g., 3.14159 =3 ad 2.71828 = 3. We always have 0 z z < 1 6