Let $\mu$ be a complex measure on the unit circle. In this thesis we study integral transforms of the measure defined on the unit disk. We investigate the behavior of its Cauchy, Poisson and conjugate Poisson transforms as the function argument approaches points lying in the support of the measure. We first show that if $\mu$ is absolutely continuous with respect to Lebesgue measure on the unit circle, the transforms tend to finite values almost everywhere on the unit circle. In contrast, if $\mu$ is singular with respect to Lebesgue measure, its Cauchy transform tends to infinity near the support of the measure. A key part of this thesis is dedicated to the normalized Cauchy transform associated with a measure $\mu$. This is a linear operator defined on the space of $\mu$-integrable functions. We discuss some properties of this operator and then turn to the main result concerning the boundary behavior of its image functions.<br />In the last chapter we present a classification of measures based on the relative speed of growth of their Poisson and conjugate Poisson transforms as their function argument approaches the support of $\mu$, and show how the results on the normalized Cauchy transform can be applied in the investigation of the boundary behavior of these transforms.
en
dc.language
English
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dc.language.iso
en
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dc.rights.uri
http://rightsstatements.org/vocab/InC/1.0/
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dc.subject
komplexe Maße
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dc.subject
Integraltransformationen
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dc.subject
Cauchy-Transformation
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dc.subject
Poisson-Transformation
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dc.subject
normalisierte Cauchy-Transformation
de
dc.subject
complex measures
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dc.subject
integral transforms
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dc.subject
Cauchy transform
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dc.subject
Poisson transform
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dc.subject
normalized Cauchy transform
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dc.title
Boundary behavior of singular integrals
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dc.type
Thesis
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dc.type
Hochschulschrift
de
dc.rights.license
In Copyright
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dc.rights.license
Urheberrechtsschutz
de
dc.contributor.affiliation
TU Wien, Österreich
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dc.rights.holder
Monika Pichler
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tuw.version
vor
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tuw.thesisinformation
Technische Universität Wien
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tuw.publication.orgunit
E101 - Institut für Analysis und Scientific Computing