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

2019/02/03 by Borichev, Alexander, Kononova, Anna, Sodin, Mikhail
#42C05 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.00874

Abstract

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space L²(ρ), where ρ is a measure on the unit circle, if and only if the logarithmic integral of the measure ρ 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 ρ on a sufficiently rare subset of the circle.

Related

• Notes on the Szego minimum problem. II. Singular measures
• On a density problem related to a theorem of Szegő
• Szegő's theorem for Jordan arcs
• Szego polynomials: a view from the Riemann-Hilbert window
• Coefficients of Orthogonal Polynomials on the Unit Circle and Higher Order Szego Theorems