Problem 4 - Fourier series Additional problems 4.1) Consider the eigenvalue problem y+y=0,y(0)=0,y(L)=0 Show that it has infinitely many positive eigenvalues given by, n=(2n1)22/4L2, with associated eigenfunctions yn=cos2L(2n1)x,n=1,2,3, 4.2) Suppose 1,2,,m are orthogonal on [a,b] and abn2dx=0,n=1,2,,m. If a1,a2,,am are arbitrary real numbers, define Pm=a11+a22++amm. Let Fm=c11+c22++cmm, where cn=abn2dxabf(x)n(x)dx that is, c1,c2,,cn are Fourier coefficients of f. 4.2a) Show that ab(f(x)Fm(x))n(x)dx=0,n=1,2,,m. 4.2b) Show that ab(f(x)Fm(x))2dx=abf2(x)dxn=1mcn2abn2dx