The periodogram IN (f ) evaluated at Fourier frequencies f = k/N is unaffected by whether or not the data are centered by the mean. Demonstrate this (mathematically) by showing that...

The periodogram IN (f ) evaluated at Fourier frequencies f = k/N is unaffected by whether or not the data are centered by the mean. Demonstrate this (mathematically) by showing that

                                   a∗ j = N−1 t=0 (yt − ¯y) exp −2πijt N "

is the same as

                               aj = N−1 t=0 yt exp −2πijt N "

for all j = 0.