2022/08/18 by Bartosz Malman, Malman, Bartosz
Mathematics · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2208.08810
openalex publication_date 2022/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the classical problem of identifying the structure of P2(μ), the closure of analytic polynomials in the Lebesgue space L2(μ) of a compactly supported Borel measure μ living in the complex plane. In his influential work, Thomson showed that the space decomposes into a full L2-space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ supported on the closed unit disk \mathbbD which have a part on the open disk \mathbbD which is similar to the Lebesgue area measure, and a part on the unit circle \mathbbT which is the restriction of the Lebesgue linear measure to a general measurable subset E of \mathbbT, we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P2(μ). It turns out that the space splits according to a certain decomposition of measurable subsets of \mathbbT which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.