Determinant calculations for the Block Szegő–Widom Limit Theorem
✍ Scribed by Torsten Ehrhardt
- Publisher
- John Wiley and Sons
- Year
- 2007
- Tongue
- English
- Weight
- 180 KB
- Volume
- 280
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
The Szegő–Widom Limit Theorem says that for certain N × N matrix functions φ defined on the unit circle, the asymptotic behavior of the determinants of the block Toeplitz matrices T~n~ (φ) is given by
equation image
where G [φ ] is the geometric mean of det φ and E [φ ] is equal to the operator determinant det T (φ)T (φ^–1^). In the scalar case (N = 1) a more explicit expression for E [φ ] exists, while in the matrix case (N > 1) not much is known.
In the present paper we are going to establish an explicit expression for E [φ ] for 2 × 2 matrix functions φ = αI + βQ where α and β are scalar functions and Q is a rational matrix function for which Q^2^ = 0. It turns out that in comparison with the scalar case, new terms in the expression for E [φ ] appear. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)