Fourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form

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1 Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier years versions of the course FXY Section XY of G Foand, Fourier Anaysis and its appications KX Chapter X of T W Körner, Fourier Anaysis T Od examinations VX Chapter X of A Vretbad, Fourier Anaysis and its appications Ö Övningsexempe i Fourieranays 003 φx ct + ψx + ct where φ and ψ are C functions of one variabe d Show that the soution of the initia vaue probem { ut t = c u xx, < x <, t > 0 ux, 0 = f x, u t x, 0 = g x, < x < is Fourier Series ux, t = 0 D39 Find the Cesàro sum of the series f x ct + f x + ct + c x+ct x ct g yd y / Introductory exercises Write e i x in terms of the trigonometric functions and cos, and conversey How are the hyperboic functions h and cosh defined? Sketch their graphs 3 Let ν > 0 The soutions of the ordinary differentia equation y ν y = 0 on the rea ine form a vector space of dimension Find a a basis of this vector space; b a basis f, g with f 0 = 0; c a basis f, g with f 0 = 0 and g = 0, where > 0 is given 4 Do the preceding exercise, but now with the equation y + ν y = 0 When is there a nontrivia soution y with y0 = y = 0, and then what about part c? 5 Let n be an integer Verify that a cosn = n ; b n + = n 6 Find a primitive of the function ax bx for nonzero a and b 7 Reca the genera soution of the differentia equation y + ay + by = 0 with constant coefficients a and b How do you sove the differentia equation y + ay + by = e βx? 8 Reca the soution method of mutipying by the integrating factor for differentia equations of the type y + g xy = 0 9 F The object of this exercise is to derive d Aembert s formua for the genera soution of the one-dimensiona wave equation u t t = c u xx a Show that if uy, z = f y + g z where f and g are C functions of one variabe, then u satisfies u yz = 0 Conversey, show that every C soution of u yz = 0 is of this form b Let y = x ct and z = x + ct Use the chain rue to show that u t t c u xx = 4c u yz c Concude that the genera C soution of the wave equation u t t = c u xx is ux, t = D39 Let a and b be rea numbers Under what conditions does a series of the form a + b + a b + a + b + a b + have a Cesàro sum? a = 0 F Sketch the graph and find the trigonometric Fourier series of the foowing functions: nt a f t = t, 0 < t < ; n b f t = t, 0 < t < 4 cosnt 4n ; c f t = t, < t < n cosnt; n 8 n t d f t = t t, < t < n 3 3 F Sketch the graph and find the exponentia Fourier series of the foowing functions: a f t = e bt hb n, < t <, b 0 n= b in eint ; b f t = e bt e b e int, 0 < t <, b 0 b in 4 F Use the resuts from exercises and 3 to show that a 4n = ; n=

2 b c d e f n+ 4n = 4 ; n = 6 ; n+ = n ; n n + b = b hb b ; n + b = b cothb b 5 K If f : T C is continuous and g = Re f show that ĝ n = the Fourier coefficients of Im f 6 F3 Starting from c, show that a t 3 n nt t =, t ; n 3 b t 4 t n+ cosnt = 48 74, t ; n 4 5 c n = F3 From b we have and we aso have t = 4 cost = d dt t = t cosnt 4n, 0 t, / sd s ˆ f n + ˆ f n Find Show that the series above can be differentiated and integrated termwise to yied two apparenty different expressions for cost for 0 < t <, and reconcie these two expressions 8 F3 Let f be the periodic function such that f t = e t for < t, and et n= c ne int be its Fouriers series; thus e t = n= c ne int for t < If we formay differentiate this equation, we obtain e t = n= inc ne int But then c n = inc n, or inc n = 0, so c n = 0 for a n This is obviousy wrong; where is the mistake? 9 F4 Express 4 n x a f x = as a e series on [0, 6]; ; n 6 b f x = x as a coe series on [0, ]; + 4 cosn x ; n c f x = x x 8 n x as a e series on [0, ]; ; 3 3 n d f x = e x as a series of the form n= c ne inx on [0, ]; e n= einx in 0 T What is the Fourier e series of the function f defined on 0, as { x, 0 < x < f x =, 0, < x <? What is the sum of the series in the points x = and x =? n+ n f has the Fourier e series x = the vaue of the series is 0, and at x = / it is /4 T Cacuate 3 f xg xd x where 0 f x = n e i n+x, and g x = n=0 n x + n+ n 0 n=0 cos4n + x * Exercises,,, 4, 9, 6, 63, 64, 73, 74 in Körner Differentia equations nx At 3 F Suppose that u and u are both soutions of the inear differentia equation Lu = f, where f 0 Under what conditions is the inear combination c u + c u aso a soution to this equation? 4 F What form must G have for the differentia equation u t t u xx = Gx, t, u to be inear? Linear and homogeneous? 5 F a Show that for n =,, 3,, u n x, y = nx hny satisfies u xx + u y y = 0, u0, y = u, y = ux, 0 = 0 b Find a inear combination of the u n s that satisfies ux, = x 3x c Show that for n =,, 3,, ũ n x, y = nx h n y satisfies u xx + u y y = 0, u0, y = u, y = ux, = 0 d Find a inear combination of the ũ n s that satisfies ux, 0 = x e Sove the Dirichet probem { uxx + u y y = 0, 0 < x, y <, u0, y = u, y = 0, 0 < y <, ux, 0 = x, ux, = x 3x, 0 < x <

3 6 F3 By separation of variabes, derive the soutions u n x, y = nx hny of { uxx + u y y = 0, 0 < x, y <, u0, y = u, y = 0, 0 < y <, ux, 0 = 0, 0 < x < from the previous exercise 7 F3 Use separation of variabes to find an infinite famiy of independent soutions of { ut = ku xx, 0 < x <, t > 0, u0, t = 0, u x, t = 0, t > 0 representing heat fow in a rod with one end hed at temperature zero and the other end insuated u n x, t = exp n+ kt n+x, n =,, 3, 4 Orthogona sets of functions and L 8 F3 Show that { / / n x/} is an orthonorma set in PC 0, 9 F3 Show that { / / cosn x/} is an orthonorma set in PC 0, 30 D3 Is, x, x, x 3, an orthogona sequence in a C [0, ]; b C [, ]? 3 F33 Show that if f n L a, b and f n f in norm, then f n, g f, g for a g L a, b 3 F33 a Show that f g f g b Deduce that if f n f in norm, then f n f 33 F34 Find constants a, b, A, B, C such that f 0 x =, f x = ax + b, and f x = Ax + Bx + C are an orthonorma set in L w 0, where wx = e x Hint: 0 x n e x dx = n! a = ; b = ; A = /; B = ; C = 34 F34 What is the best approximation in norm fo the function f x = x on the interva [0, ] among a functions of the form a a 0 + a cos x + a cos x; 4 cos x; b b x + b x; x x; c a cos x + b x? 4 cos x + x einx are pairwise orthogona in L R Find the 35 Ö Show that the functions φ n x = x x numbers c n that minimizes the integra N + x c n φ n x dx n= N c n = e e e n if n 0 and c 0 = e 36 Ö Find the soution yx to y y = 0 that minimizes + x yx dx yx = h h + cosh x + e h x h 37* Exercises 7, 7, 33, 343 in Körner Sturm-Liouvie probems 38 F35 Find the eigenvaues and normaized eigenfunctions for the probem f + λf = 0, f 0 = 0, f = 0 on [0, ] The eigenvaues are λ n = n /, with eigenfunctions / / n /x/, n =,, 3, 39 F35 Find the normaized eigenfunctions for the probem f + λf = 0, f 0 = 0, f = βf Note that the answers wi be a bit different in the three cases β < 0, β = 0 and β > 0 If β < 0, the eigenvaues are the numbers λ n = ν n where the ν n s are the positive soutions of tan ν = β/ν, and the corresponding normaized eigenfunctions are β φ n x = ν n β cos ν nx If β = 0, the eifenvaues are λ n = n/, ie, the squares of the soutions of tan ν = 0, incuding λ 0 = 0 The normaized eigenfunctions are φ n x = / / cosnx/, n > 0 and φ 0 x = / If β > 0, the eigenvaues are the squares of the positive soutions ν n of the equation tan ν = β/ν, together with the square of the unique positive soution µ 0 of tanh µ = β/µ The normaized eigenfunctions are β φ n x = β ν n cos ν β nx, φ 0 x = β + h µ 0 cosh µ 0x The foowing probem uses the fact that the genera soution of the Euer equation x f x + ax f x + b f x = 0, x > 0 is c x r + c x r where r and r are the zeros of the poynomia r r + ar + b If two zeros coincide, the genera soution is c x r + c x r og x In case r and r are compex, it is usefu to reca that x i s = e i s og x 40 F35 Find the eigenvaues and normaized eigenfunctions for the probem

4 x f + λx f = 0, f = f b = 0, x b, b > Expand the function g x = in terms of these eigenfunctions Hint: in computing integras, make the substition y = og x Orthonormaity here is with respect to the weight wx = x λ n = n/ og b, φ n x = g x = 4 n og b n og x og b n og x og b 4 Ö Find a eigenvaues and corresponding eigenfunctions of the Sturm-Liouvie probem e 4x e 4x u = λu, 0 < x <, u0 = 0, u = 0 Express e x in a series of the eigenfunctions λ = r β, where β is the positive soution to tanh β = β ; u x = e x h β x; λ n = 4 + β n, where β n, n =, 3, are the positive soutions to the equation tan β = β ; u nx = e x β n x The function e x has the series expansion λ n λ n + n β n λ n u n x 4 EÖ Consider the Sturm-Liouvie operator L f = r f + q f + p f, where the coefficients are rea functions with r C, q C and p C 0 in the interva [a, b] Assume that L is formay sef-adjoint in the sense that L f, g = f, Lg for a f, g C [a, b] such that the functions f, f, g, g are a zero at a and b Prove that then q = r, so that L is of the form L f = r f + p f 43* Exercise 395 in Körner b What if f x 50? { ut = ku xx, 0 < x <, t > 0, ux, 0 = f x, 0 < x <, u0, t = u x, t = 0, t > 0 a Let b n = n x f x dx Then 0 ux, t = b n exp n kt n x 4 b b n = 00/n 45 F4 Repeat 44a but repacing u0, t = 0 with the assumption u0, t = C 0 With b n as in the answer of 44a, 4C ux, t = C + b n exp n kt n x n 4 46 F4 Repeat 44a, assuming that the rod generates heat within itsef at a constant rate R, so the heat equation is repaced by u t = ku xx + R With b n as in the answer of 44a, ux, t = R x x k + 6R b n exp n kt k 3 n F4 What if the heat equation is repaced by u t = ku xx + Re ct? With b n as in the answer of 44a, ux, t = 4R n odd e ct e n kt/ nn k/ c n x nx, if c n k/ for any odd n If c = N k/ for some odd N, the coefficient of Nx is N te N kt/ 48 T Sove the PDE { ut = u xx + x, 0 < x <, t > 0 u0, t = u, t = 0, t > 0, ux, 0 =, 0 < x < More differentia equations 44 F4 a Find a series expansion for the temperature ux, t in a rod of ength that is hed at temperature zero at one end and the other end insuated, with initia temperature f x Such a rod satisfies the onedimensiona heat equation ux, t = + 4 e t x t n e n n x n=

5 49 T Sove the PDE { ut t = u xx, 0 < x <, t > 0 u0, t =, u, t =, t > 0, ux, 0 =, u t x, 0 =, 0 < x < ux, y = c n e ny/ nx, c n = 0 f x nx dx ux, t = x + 4 n+ n x cos n t n 8 n n x t n 50 F43 Find the genera soution in terms of a series of u t t = c u xx a u, u0, t = u, t = 0, with arbitrary initia conditions This is a mode for a string vibrating in an eastic medium ux, t = an cos λ n t + b n λ n t nx, λ n = nc + a 5 F43 A string of ength = is fixed at one and and attached to an osciator at the other, so that u0, t = 0 and u, t = kt If the string is initiay at rest ux, 0 = u t x, 0 = 0, find ux, t Hint: Let vx, t = ux, t x/ kt and sove for vx, t ux, t = x kt + k n+ nn c k k kt nc nct nx If k = Nc, the N th term of the series is N Nct + Nct cos Nct N x/n c 5 F44 Sove the boundary vaue probem ux, y = y 4 { uxx + u y y = 0, 0 < x, y <, u x 0, y = u x, y = 0, 0 < y <, ux, 0 = 0, ux, = x, 0 < x < n hn cos n x 53 F44 Express the bounded soution to the probem u xx + u y y = 0, 0 < x <, 0 < y <, u0, y = u, y = 0, 0 < y <, ux, 0 = f x, 0 < x < as a series n y h 54 Ö Sove the probem { uxx + u y y = y, 0 < x <, 0 < y <, ux, 0 = ux, = 0, 0 < x <, u0, y = y y 3, u, y = 0, 0 < y < ux, y = 6 y 3 y + 3 Convoutions, Fourier transforms n n 3 h n h nx + 7 h n x ny 55 F7 Let f x = x p, where / < p < Show that f is neither in L nor L, but that f can be expressed as a sum of an L function and an L function 56 F7 Let f x = e x and g x = e x Compute f g Hint: Compete the square in the exponent and use the fact that e x dx = f g x = /3e x /3 57 V7 A function f is defined as { x, x <, f x = 0, x > Find the Fourier transform f ˆ of f f ˆ ξ = 4i ξ cos ξ ξ/ξ 58 V7 Compute the Fourier transform of { x x < f x = 0 x > Use the resut to compute 4 ξ dξ ξ 4 f ˆ ξ = 4 ξ and the integra is /3 ξ 59 Ö Compute the Fourier transform of a, b > 0 x i a ; x + a a ξe a ξ ; b x + a ; + a ξ e a ξ ; a 3 c e a x 4i abξ bx; ξ +bξ+a +b ξ bξ+a +b 60 Ö The function f x has Fourier transform f ˆ ξ = ξ/ + ξ 4 Show that

6 a x f xdx = i ; b f 0 = i 6 Ö The function f x has Fourier transform f ˆ ξ = / + ξ 3 Show that f f dx = 9 6 F7 For t > 0, set f t x = 4t / e x /4t Show that f t f s = f t+s 63 F7 For a > 0, et f a x = a/ x + a and g a x = ax/x Us e the Fourier transform to show that a f a f b = f a+b ; b g a g b = g mina,b Hint: F /x + a ξ = /ae a ξ and F ax/x ξ = χ a ξ where χ a ξ is equa to one if ξ < a and zero esewhere 64 F7 Let a, b > 0 Use the Panchere theorem to show ax bx a dx = mina, b; b x x x + a x + b dx = a + b 65 F73 Use the Fourier transform to find a soution of the ordinary differentia equation u u + g x = 0 where g L The soution obtained this way is the one that vanishes at ± What is the genera soution? ux = g e x = e x x e y g yd y + e x x e y g yd y 66 F73 Consider the wave equation u t t = c u xx with initia conditions ux, 0 = f x and u t x, 0 = g x a Assuming that a the Fourier transforms in question exist, show that Lapace transforms 69 F8 Cacuate the Lapace transform of f t = cos t L f z = z + zz F8 Let a, b > 0 and a b Cacuate f g by ug Lapace transform, if a f t = e at, g t = e bt e at e bt ; a b ; b f t = t, g t = t 7 F83 Sove the foowing initia vaue probems ug the Lapace transform a u + 4u = ωt, u0 = u 0 = 0, ω > 0 b { u = u u + e 4t, u 0 = 3 u = 3u + u 3e 4t, u 0 = 8 3 t 3 t; a If ω then ut = ω t ωt, if ω = then ut = ω 4 8 t t cos t b u t = 5e t e 4t + te 4t and u t = 5e t + 3e 4t 3te 4t ûξ, t = ˆ f ξ cosctξ + ĝ ξ ctξ b Invert the Fourier transform to obtain d Aembert s formua for u 67 F73 Suppose that f L represents a signa Show that the best approximation to f in the L norm among a signas that are band-imited to the interva [ Ω, Ω] is g 0 t = Ω f ˆ ωdω That is, show that g Ω 0 f g f for a g such that ĝ ω = 0 for ω > Ω 68* Exercise 46, 47, 5, 55, 55, 606, 607, 609 in Körner