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dc.contributor.advisorBoas, Harold P.en_US
dc.creatorJacobson, Robert Lawrenceen_US
dc.date.accessioned2012-07-16T15:58:27Zen_US
dc.date.accessioned2012-07-16T20:24:44Z
dc.date.available2012-07-16T15:58:27Zen_US
dc.date.available2012-07-16T20:24:44Z
dc.date.created2012-05en_US
dc.date.issued2012-07-16en_US
dc.date.submittedMay 2012en_US
dc.identifier.urihttp://hdl.handle.net/1969.1/ETD-TAMU-2012-05-11068en_US
dc.description.abstractThe classical Lu Qi-keng Conjecture asks whether the Bergman kernel function for every domain is zero free. The answer is no, and several counterexamples exist in the literature. However, the more general Lu Qi-keng Problem, that of determining which domains in Cn have vanishing kernels, remains a difficult open problem in several complex variables. A challenge in studying the Lu Qi-keng Problem is that concrete formulas for kernels are generally difficult or impossible to compute. Our primary focus is on developing methods of computing concrete formulas in order to study the Lu Qi-keng Problem. The kernel for the annulus was historically the first counterexample to the Lu Qi-keng Conjecture. We locate the zeros of the kernel for the annulus more precisely than previous authors. We develop a theory giving a formula for the weighted kernel on a general planar domain with weight the modulus squared of a meromorphic function. A consequence of this theory is a technique for computing explicit, closed-form formulas for such kernels where the weight is associated to a meromorphic kernel with a finite number of zeros on the domain. For kernels associated to meromorphic functions with an arbitrary number of zeros on the domain, we obtain a weighted version of the classical Ramadanov's Theorem which says that for a sequence of nested bounded domains exhausting a limiting domain, the sequence of associated kernels converges to the kernel associated to the limiting domain. The relationship between the zeros of the weighted kernels and the zeros of the corresponding unweighted kernels is investigated, and since these weighted kernels are related to unweighted kernels in C^2, this investigation contributes to the study of the Lu Qi-keng Problem. This theory provides a much easier technique for computing certain weighted kernels than classical techniques and provides a unifying explanation of many previously known kernel formulas. We also present and explore a generalization of the Lu Qi-keng Problem.en_US
dc.format.mimetypeapplication/pdfen_US
dc.language.isoen_USen_US
dc.subjectseveral complex variablesen_US
dc.subjectcomplex analysisen_US
dc.subjectBergman kernelen_US
dc.titleWeighted Bergman Kernel Functions and the Lu Qi-keng Problemen_US
dc.typeThesisen
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorTexas A&M Universityen_US
thesis.degree.nameDoctor of Philosophyen_US
thesis.degree.levelDoctoralen_US
dc.contributor.committeeMemberSmith, Rogeren_US
dc.contributor.committeeMemberStraube, Emil J.en_US
dc.contributor.committeeMemberCline, Daren B.en_US
dc.type.genrethesisen_US
dc.type.materialtexten_US


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