On Approximate Spectral Factorization of Matrix Functions |
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Authors: | Lasha Ephremidze Gigla Janashia Edem Lagvilava |
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Institution: | 1. I. Javakhishvili State University, 2, University Street, Tbilisi, 0143, Georgia 2. A. Razmadze Mathematical Institute, 1, M. Aleksidze Str., Tbilisi, 0193, Georgia
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Abstract: | It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S
n
, n=1,2,… , are convergent in the L
1-norm, ||Sn-S||L1? 0\|S_{n}-S\|_{L_{1}}\to 0, and ò02plogdetSn(eiq) dq?ò02plogdetS(eiq) dq\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta, then the corresponding (canonical) spectral factors are convergent in L
2, ||S+n-S+||L2? 0\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place. |
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Keywords: | |
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