On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian $n\times n$ block Toeplitz matrices $A_{n,m}$, with $m\times m$ Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function $f$, and we study their eigenvalues for large $n$ and $m$, relating their behaviour to some properties of $f$ as a function; in particular we show that, for any fixed $k$, the first $k$ eigenvalues of $A_{n,m}$ tend to $\inf f$, while the last $k$ tend to $\sup f$, so extending to the block case a well-known result due to Szegö. In the case the $A_{n,m}$'s are positive-definite, we study the asymptotic spectrum of $P_{n,m}^{-1}A_{n,m}$, where $P_{n,m}$ is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system $A_{n,m}x=b$, obtaining strict estimates, when $n$ and $m$ are fixed, and exact limit values, when $n$ and $m$ tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.