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Journal of Fourier Analysis and Applications

, Volume 17, Issue 5, pp 976–990 | Cite as

On Approximate Spectral Factorization of Matrix Functions

  • Lasha Ephremidze
  • Gigla Janashia
  • Edem Lagvilava
Article

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, \(\|S_{n}-S\|_{L_{1}}\to 0\), and \(\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta\), then the corresponding (canonical) spectral factors are convergent in L 2, \(\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0\). The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place.

Keywords

Matrix spectral factorization 

Mathematics Subject Classification (2010)

47A68 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Lasha Ephremidze
    • 1
    • 2
  • Gigla Janashia
    • 2
  • Edem Lagvilava
    • 2
  1. 1.I. Javakhishvili State UniversityTbilisiGeorgia
  2. 2.A. Razmadze Mathematical InstituteTbilisiGeorgia

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