fourier seriesa fourier series may be truncated

Question

Fourier series

A Fourier series may be truncated to the form

For each of the following functions nd the Fourier coecients in their simplest form and hence nd their Fourier Series representation. On the same graph plot the function and the corresponding truncated Fourier series for the given values of N. In all cases the period used is -T/2

(i) f(x) = x³, N = 2, 4, 8, 16

(ii) f(x) = (1 x²), N = 2, 4, 8, 16

(iii) f(x) = (1 x)² (1 + x)², N = 2, 4, 8

(iv) f(x) = x(u(x) u(x 1)) , N = 2, 4, 8, 16

Use the graphs and the formulae for the Fourier coecients to discuss the convergence rates. You should consider the behaviour of the functions at their end and midpoints points to suggest features that increase the convergence and those that are bad for convergence.