Question

An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular, show that Ln is orthogonal to zwhenever m

Answer 1

0 Soluti on- We no thal In(A) Satiafied the Legendric diffezn uLiping thi en igyating wy.t -r betwee n 61 . we have

PnkeyaLina L.H. pants we have, 1-F -1 The first teymin L.H.づ vanishes en Substi LuLione mnow, we have Cnmiloriy e ao oblain We have
,''''''''''''''''''''jjj'''np[
Cmim+-n(fin.di Thus f m gince, mnowhich mplies that