Distribution results on the algebra generated by Toeplitz sequences: a finite-dimensional approach

Let a be an L-1 symbol defined on (Q(d), Q = (-pi, pi) with d greater than or equal to 1 and let us consider the multi-indexed sequence of Toeplitz ma trices {T-n(a)} with n is an element of N-d. It is well known that {T-n(a)} is spectrally distributed as a in the sense of the singular values. In thi s paper we prove that a sequence as {Sigma (alpha) Pi T-beta(n)(a(alpha bet a))} is spectrally distributed in the sense of the singular values as the m easurable function theta = Sigma (alpha) Pi (beta)a(alpha beta) for a(alpha beta) is an element of L-1 and alpha and beta ranging in any finite set of values. (C) 2001 Elsevier Science Inc. All rights reserved.