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On determination of the optimal factor of a nonnegative matrix-valued function


Author: Habib Salehi
Journal: Proc. Amer. Math. Soc. 29 (1971), 383-389
MSC: Primary 46.30; Secondary 47.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0278056-0
MathSciNet review: 0278056
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Abstract: Let $ F = [{f_{ij}}],1 < i,j \leqq q$, be a measurable, nonnegative definite $ q \times q$ matrix-valued function defined on the unit circle C. It is known that when F and $ \log\, \det \textbf{F}$ are in $ {L_1}(C)$, F admits a factorization of the form $ F = {\mathbf{\Phi }}{{\mathbf{\Phi }}^ \ast }$, where $ {\mathbf{\Phi }}$ is an optimal, full rank function in $ L_2^{0 + }(C)$. Under the additional assumption that $ \{ (\prod\nolimits_{i = 1}^q {{f_{ii}})/\det F\} } $ is in $ {L_1}(C)$, an iterative procedure which yields an infinite series for $ {\mathbf{\Phi }}$ in terms of F is given. The optimal function $ {\mathbf{\Phi }}$ plays a significant role in the multivariate prediction theory of stochastic processes. The present work generalizes the results of several authors concerning the determination of the optimal factor $ {\mathbf{\Phi }}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0278056-0
Keywords: Factorization problem, optimal factor, prediction theory, iterative procedure, eigenvalue, determinant, Fourier coefficient, bounded operator
Article copyright: © Copyright 1971 American Mathematical Society

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