Parseveral Theorem

Theorem 4: Parseval’s theorem

If $f$ is $L^2[-\tau /2,\tau /2]$ and $\tau$ periodic function then $\frac{1}{\tau }{\int }_{-\tau /2}^{\tau /2}|f(t){|}^{2}dt={\sum }_{n=-\infty }^{\infty }|c_n{|}^2$

Theorem 5: Parseval’s theorem

If $f$ is a real valued PWC1 and τ periodic function then $1/\tau {\int }_{-\tau /2}^{-\tau /2}|f(t){|}^{2}dt=c_0^2+\frac{1}{2}{\sum }_{n=1}^{\infty }(c_n^2+s_n^2)$

where $c_n$ and $s_n$ are the Fourier cosine and sine coefficients.