Abstract
An elementary and direct proof of the Szegö formula is given, for both eigen and singular values. This proof, which is based on tools from linear algebra and does not rely on the theory of Fourier series, simultaneously embraces multilevel Toeplitz matrices, block Toeplitz matrices and combinations of them. The assumptions on the generating function f are as weak as possible; indeedf is a matrix-valued function of p variables, and it is only supposed to be integrable. In the case of singular values f(x), and hence the block p-level Toeplitz matrices it generates, are not even supposed to be square matrices. Moreover, in the asymptotic formulas for eigen and singular values the test functions involved are not required to have compact support.
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