Notes on the Szego minimum problem. I. Measures with deep zeroes

Alexander Borichev; Anna Kononova; Mikhail Sodin

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arXiv · 1902.00874

Notes on the Szego minimum problem. I. Measures with deep zeroes

Alexander Borichev·Anna Kononova·Mikhail Sodin

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Abstract

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In this note we give a quantitative version of Szego's theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of the circle.

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Alexander Borichev, Anna Kononova, Mikhail Sodin. 2019-02-03. Notes on the Szego minimum problem. I. Measures with deep zeroes. https://arxiv.org/abs/1902.00874

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