Hi just wondering if
N N/3
∑ y(n) * e-J2πfn = ∑ y(n) * e-J2πfn } ?
(2*N/3) 0
Do you mean, if
$$\sum_{n=0}^{N/3} y_n e^{-2 i \pi n f} = \sum_{n = 2 N / 3}^{N} y_n e^{-2 i \pi n f}$$
or, in other words, if the Fourier series sum at \$f\$ has symmetric contributions from \$n = 0 \dots N/3\$ and \$n = 2N/3 \dots N\$?
The answer is generally no. (There are some specific configurations of \$y_n\$ for which it is true, but they tend to be uninteresting.)