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We study the asymptotic behaviour of the eigenvalues of Hermit- ian n n block Toeplitz matrices An;m ,w ith m mToeplitz blocks. Such matrices are generated by the Fourier coecients 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 xed k, the rst k eigenvalues of An;m tend to inf f, while the last k tend to supf, so extending to the block case a well-known result due to Szego. In the case the An;m's are positive-denite, we study the asymptotic spectrum of P 1 n;mAn;m ,w here Pn;m is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system An;mx = b, obtaining strict estimates, when n and m are xed, and exact limit values, when n and m tend to innity, 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.

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