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

Let a be an L1 symbol defined on Qd, Q=(-π,π) with d≥1 and let us consider the multi-indexed sequence of Toeplitz matrices {Tn(a)} with n∈Nd. It is well known that {Tn(a)} is spectrally distributed as a in the sense of the singular values. In this paper we prove that a sequence as {∑α∏βTn(a αβ)} is spectrally distributed in the sense of the singular values as the measurable function θ=∑α∏βaαβ for aαβ∈L1 and α and β ranging in any finite set of values.