L([0,2]) = (f: [0,2] C| |||2 exists in R>0} where the norm |||2: 1([0, 2])+...

  

Transcribed Image Text:

L²([0,2]) = (f: [0,2] → C| |||2 exists in R>0} where the norm |||2: 1²([0, 2])+ R20 and the sesquilinear form (,): L²([0, 2]) x L²([0, 2]) → C are given by (1.9) = Define 1(0)9(0)4dt and f(t)g(t)dt and Let f2= √(f.f). em (t) = eint, for m € Z and t€ R₁0,2] d Let L: L²(R,2]) → L² (R₁0,2x)) be the operator on L²(R02]) given by L= Let me Z. Show that em is an eigenvector of L and compute the eigenvalue. Show that (eo, e₁,e-1,e2,e-2, ...) is an orthonormal sequence in L²([0,2]). Determine aq, a1, a-1, a2, a-2,... E C such that t = ageo+aje1+a-je-1 +aze2 + a_2e-2+. Determine co, C₁, C-1,C2,c-2,... E C such that t² = coeo + ciei +c-1e-1+c₂e2+c-2e-2+... L²([0,2]) = (f: [0,2] → C| |||2 exists in R>0} where the norm |||2: 1²([0, 2])+ R20 and the sesquilinear form (,): L²([0, 2]) x L²([0, 2]) → C are given by (1.9) = Define 1(0)9(0)4dt and f(t)g(t)dt and Let f2= √(f.f). em (t) = eint, for m € Z and t€ R₁0,2] d Let L: L²(R,2]) → L² (R₁0,2x)) be the operator on L²(R02]) given by L= Let me Z. Show that em is an eigenvector of L and compute the eigenvalue. Show that (eo, e₁,e-1,e2,e-2, ...) is an orthonormal sequence in L²([0,2]). Determine aq, a1, a-1, a2, a-2,... E C such that t = ageo+aje1+a-je-1 +aze2 + a_2e-2+. Determine co, C₁, C-1,C2,c-2,... E C such that t² = coeo + ciei +c-1e-1+c₂e2+c-2e-2+...

Answer rating: 100% (QA)

To show that eimt is an eigenvector of L we need to show that Leimt lambda eimt for some scalar lamb... View the full answer