Question

*Prove that the orthogonal systems {sin nx)., and {ï, cos nxml are both complete on [0, T] 3.

Answer 1

a_n {sin nx} b_0 = 1, b_n = {cos n x} the system {a_n, b_n} spans the functions space in [0, pi] because each function has fourier representation f(x) = Sum_n = 0^infinity b_n cos n pi x/l + Sum_n = 1^infinity a_n sin n pi x/l Here l = pi Therefore f(x) = Sum_n = 0^infinity b_n cos nx + Sum_n = 1^infinity a_n sin nx and < f(x), cos nx > = 0 n = 0, 1, 2 ..... and < f(x), cos nx > = integral_0^pi f(x) cos nx dx < f(x), sin nx > = integral_0^pi f(x) sin nx dx < f(x), sin nx > = 0 n = 1, 2 ..... implies a_n = 0 and b_n = 0 Therefore f(x) = 0 Hence {a_n, b_n} is 2 basis. Hence {a_n, b_n} spans the space and {a_n, b_n} is a basis. Hence {a_n, b_n} both complete on [0, pi]