What is the inverse DTFT of the $2\pi$-periodic extension of following function:
$$H_1(\Omega)=\begin{cases} 10,& \text{for } \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& \text{for } 0 \leq |\Omega| < \frac{\pi}{3}\\ \end{cases}$$
I have found out using the definition that it is: $$h[n]=-\frac{10\sin(\frac{\pi}{3}n)}{\pi n}$$
But the problem is that this has a DTFT of
$$H_2(\Omega)=\begin{cases} -10,& \text{for } 0 \leq |\Omega| \leq \frac{\pi}{3}\\ 0,& \text{for } \frac{\pi}{3} < |\Omega| < \pi\\ \end{cases}$$
Now, $H_1(\Omega)$ and $H_2(\Omega)$ seem quite different to me. Am I missing something? Have I done something wrong in my calculations?