A matrix generalization of a theorem of SzegŐ
Copyright policy )A matrix generalization of a theorem of SzegŐ
Let \(\lambda\) be the normalized Lebesgue measure on the unit circle \(\mathbb{T}\) of the complex plane. For a non-negative finite Borel measure \(\mu\) on \(\mathbb{T}\), let \(\mu'\) denote the Radon-Nikodým derivative of the absolutely continuous part of \(\mu\) with respect to \(\lambda\). A well- known theorem of Szegö, which has important applications in prediction theory, states that for \(p\in (0,\infty)\) \[ \exp \int_{\mathbb{T}} \log\mu' d\lambda=\inf \int_{\mathbb{T}} | 1-t|^ p d\lambda \] with the convention \(\exp\int_{\mathbb{T}} \log\mu' d\lambda= 0\) if \(\log\mu'\) is not integrable. Here \(t\) runs through the set of polynomials such that \(t(0)= 0\). \textit{V. N. Zasuhin} [C. R. (Dokl.) Akad. Sci. URSS, N. Sér. 33, 435-437 (1941)] announced and \textit{H. Helson} and \textit{D. Lowdenslager} [Acta Math. 99, 165-202 (1958; Zbl 0082.282)] proved a matrix version of Szegö's theorem in the case \(p=2\). In our paper we extend the result of these authors to arbitrary \(p\in (0,\infty)\).
- Leipzig University Germany
Trigonometric approximation, \(L^ p\)-spaces, Fejér means, Radon-Nikodým derivative, Prediction theory (aspects of stochastic processes), Szegö's theorem, non-negative Hermitian matrix-valued measures, Fourier series, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Trigonometric approximation, \(L^ p\)-spaces, Fejér means, Radon-Nikodým derivative, Prediction theory (aspects of stochastic processes), Szegö's theorem, non-negative Hermitian matrix-valued measures, Fourier series, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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