I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation.
Let $X_n$ be an $n \times n$ Toeplitz matrix with associated function $f(\omega)$ where $f(\omega)$ is band limited, i.e., $f(\omega)=0$ for $|\omega|>\beta \pi$, for some $\beta < 1$. Thus, element (k,l) in $X_n$ is $$\int_{-\pi}^{\pi}f(\omega)\exp(-(k-l)i \omega)d\omega$$.
Is $X_n$ invertible as $n\to \infty$ ?
For any finite $n$ it is, but to me it seems as Szegö's theorem, which gives the asymptotic eigenvalue distribution, will kick so the the inverse does not exist as $n \to \infty $.&=
Using a particular construction, which requires $X_n$ to be invertible, the authors can prove some remarkable things, as $n\to \infty$. I believe that the mistake is exactly that they assume that $X_n$ is invertible as $n\to \infty$….