Radial limits of holomorphic functions on the ball

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dc.contributor.advisor Boas , Harold P . en_US
dc.contributor.committeeMember Cline , Daren B . H . en_US
dc.creator Fulkerson , Michael C en_US
dc.date.accessioned 2008 -10 -10T21 :01 :03Z
dc.date.accessioned 2014 -02 -19T19 :29 :42Z
dc.date.available 2008 -10 -10T21 :01 :03Z
dc.date.available 2014 -02 -19T19 :29 :42Z
dc.date.created 2008 -08 en_US
dc.date.issued 2008 -10 -10T21 :01 :03Z
dc.identifier.uri http : / /hdl .handle .net /1969 .1 /86034
dc.description.abstract In this dissertation , we consider various aspects of the boundary behavior of holomorphic functions of several complex variables . In dimension one , a characterization of the radial limit zero sets of nonconstant holomorphic functions on the disc has been given by Lusin , Privalov , McMillan , and Berman . In higher dimensions , no such characterization is known for holomorphic functions on the unit ball B . Rudin posed the question as to the existence of nonconstant holomorphic functions on the ball with radial limit zero almost everywhere . Hakim , Sibony , and Dupain showed that such functions exist . Because the characterization in dimension one involves both Lebesgue measure and Baire category , it is natural to also ask whether there exist nonconstant holomorphic functions on the ball having residual radial limit zero sets . We show here that such functions exist . We also prove a higher dimensional version of the Lusin -Privalov Radial Uniqueness Theorem , but we show that , in contrast to what is the case in dimension one , the converse does not hold . We show that any characterization of radial limit zero sets on the ball must take into account the "complex structure" on the ball by giving an example that shows that the family of these sets is not closed under orthogonal transformations of the underlying real coordinates . In dimension one , using the theorem of McMillan and Berman , it is easy to see that radial limit zero sets are not closed under unions (even finite unions ) . Since there is no analogous result in higher dimensions of the McMillan and Berman result , it is not obvious whether the radial limit zero sets in higher dimensions are closed under finite unions . However , we show that , as is the case in dimension one , these sets are not closed under finite unions . Finally , we show that there are smooth curves of finite length in S that are non -tangential limit uniqueness sets for holomorphic functions on B . This strengthens a result of M . Tsuji . en_US
dc.format.medium electronic en_US
dc.language.iso en _US en_US
dc.publisher Texas A &M University en_US
dc.subject radial limits en_US
dc.title Radial limits of holomorphic functions on the ball en_US
dc.type Book en
dc.type.genre Electronic Dissertation en_US
dc.type.material text en_US
dc.format.digitalOrigin born digital en_US

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Radial limits of holomorphic functions on the ball. Available electronically from http : / /hdl .handle .net /1969 .1 /86034 .

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