On the direct cauchy theorem in widom domains: Positive and negative examples

Peter Yuditskii

doi:10.1007/bf03321869

On the direct cauchy theorem in widom domains: Positive and negative examples

Peter Yuditskii

Johannes Kepler University Linz

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

We discuss several questions which remained open in our joint work with M. Sodin Almost periodic Jacobi matrices with homogeneous spectrum, in nite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. In particular, we show that there exists a nonhomogeneous set E such that the Direct Cauchy Theorem (DCT) holds in the Widom domain C\E. On the other hand we demonstrate that the weak homogeneity condition on E (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.

Original language	English
Pages (from-to)	395-414
Number of pages	20
Journal	Computational Methods and Function Theory
Volume	11
Issue number	2
DOIs	https://doi.org/10.1007/bf03321869
State	Published - Jan 2012
Externally published	Yes

Bibliographical note

Funding Information:
Supported by the Austrian Science Fund FWF, project no: P22025-N18

Funding

Supported by the Austrian Science Fund FWF, project no: P22025-N18

Funders	Funder number
Austrian Science Fund	P22025-N18

Keywords

Direct Cauchy Theorem
Entire functions
Hardy spaces on Riemann surfaces
Homogeneous set
Martin function
Reflectionless Jacobi matrices
Widom domain

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10.1007/bf03321869

Cite this

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abstract = "We discuss several questions which remained open in our joint work with M. Sodin Almost periodic Jacobi matrices with homogeneous spectrum, in nite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. In particular, we show that there exists a nonhomogeneous set E such that the Direct Cauchy Theorem (DCT) holds in the Widom domain C\E. On the other hand we demonstrate that the weak homogeneity condition on E (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.",

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AB - We discuss several questions which remained open in our joint work with M. Sodin Almost periodic Jacobi matrices with homogeneous spectrum, in nite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. In particular, we show that there exists a nonhomogeneous set E such that the Direct Cauchy Theorem (DCT) holds in the Widom domain C\E. On the other hand we demonstrate that the weak homogeneity condition on E (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.

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On the direct cauchy theorem in widom domains: Positive and negative examples

Abstract

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Keywords

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