ON THE ASYMPTOTIC SPECTRUM OF HERMITIAN BLOCK TOEPLITZ MATRICES WITH TOEPLITZ BLOCKS

We study the asymptotic behaviour of the eigenvalues of Hermitian n x n block Toeplitz matrices A(n,m), with m x m Toeplitz blocks. Such mat rices are generated by the Fourier coefficients of an integrable bivar iate function f, and we study their eigenvalues for large n and m, rel ating their behaviour to some properties of f as a function; in partic ular 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 fr so extending to the blo ck case a well-known result due to Szego. 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,P-m is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system A(n,m)x = b, Obtaining s trict 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 conjug ate gradient convergence factor of the previous matrices. Extensions t o the case of a deeper nesting level of the block structure are also d iscussed.