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A very short proof of the Fejér-Riesz lemma is presented in the matrix case. The following fundamental result in matrix spectral factorization theory belongs to Wiener [8] (see also [5], [4]): Theorem. Let (1) S(z) ∼ ∞ n=−∞ σ n z n , |z| = 1, σ k are r×r matrix coefficients, be a positive definite matrix-function with integrable entries, S(z) ∈ L 1 (T). If(More)
An effective factorization and partial indices are found for a class of unitary matrix functions. Let R denote a normed ring of functions defined on the unit circle of a complex plane, say, a ring H α of Hölder functions with a usual norm, 0 < α < 1, which can be decomposed into the direct sum of its subrings R = R + + R − 0 , where the elements R + are the(More)
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