Question

4) (5 marks) Find a smooth function f(x) for x є [-1, 1] that cannot be represented by (TE) n=1 for any values of the constants ao and an. Explain the reasons for your answer

Answer 1

Fourier series of f (x) in [- l, l] f(x) = a_0/2 + sigma_n = 1^infinity [a_n cos(n pi/l x) + an sin (n pi/l x)] Here [- l, l] = [- 1, 1] therefore l = 1 if f(x) is even function then b_n = 0 and f(x) is odd function then a_0, a_n = 0 so f(x) = x then f(x) is odd function in [- 1,1] in this case f(x) cannot be represented by a_0 + sigma n = 1^infinity a_n cos (n pi x) and it can be represented by sigma_n = 1^infinity b_n sin(n pi x) only therefore f(x) = x