Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded Toeplitz matrices $T_n(H)$. The Grenander-Szegö theorem states that the spectrum of $T_n(H)$ converges to $H(\omega)$ as $n\to \infty$.
My question is: For finite $n$, is $T_n(H)$ always strictly positive definite ?
If it simplifies, one can assume that $H(\omega)=1, \, |\omega|<\beta \pi$, i.e., a "brickwall" spectrum.
