Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains vii, 79 p. Includes bibliographical references (p. 79). Available online via OhioLINK's ETD Center

The thesis consists of three articles concerning reproducing kernels ofweighted spaces of polyanalytic functions on the complex plane. In the first paper, we study spaces of polyanalytic polynomials equipped with a Gaussianweight. In the remaining two papers, more general weight functions are considered. More precisely, we provide two methods to compute asymptotic expansions for the kernels near the diagonal and then apply the techniques to get estimates for reproducing kernels of polyanalytic polynomial spaces equipped with rather general weight functions. / <p>QC 20130513</p>

The 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.

Thesis (doctoral)--Universität Bonn, 2003. / Includes bibliographical references (p. 163-165).

Let (L,h) → (X,ω) be a compact toric polarized Kahler manifold of complex dimension n. For each k ε N, the fibre-wise Hermitian metric hk on Lk induces a natural inner product on the vector space C∞(X,Lk) of smooth global sections of Lk by integration with respect to the volume form ωn /n! . The orthogonal projection Pk : C∞(X,Lk) → H0(X,Lk) onto the space H0(X,Lk) of global holomorphic sections of Lk is represented by an integral kernel Bk which is called the Bergman kernel (with parameter k ε N). The restriction ρk : X → R of the norm of Bk to the diagonal in X × X is called the density function of Bk. On a dense subset of X, we describe a method for computing the coefficients of the asymptotic expansion of ρk as k → ∞ in this toric setting. We also provide a direct proof of a result which illuminates the off-diagonal decay behaviour of toric Bergman kernels. We fix a parameter l ε N and consider the projection Pl,k from C∞(X,Lk) onto those global holomorphic sections of Lk that vanish to order at least lk along some toric submanifold of X. There exists an associated toric partial Bergman kernel Bl,k giving rise to a toric partial density function ρl,k : X → R. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of X as k → ∞. Euler-Maclaurin sums and Laplace’s method are utilized as important tools for this. We discuss the case of a polarization of CPn in detail and also investigate the non-compact Bargmann-Fock model with imposed vanishing at the origin. We then discuss the relationship between the slope inequality and the asymptotics of Bergman kernels with vanishing and study how a version of Song and Zelditch’s toric localization of sums result generalizes to arbitrary polarized Kahler manifolds. Finally, we construct families of induced metrics on blow-ups of polarized Kahler manifolds. We relate those metrics to partial density functions and study their properties for a specific blow-up of Cn and CPn in more detail.

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Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 75-80).

Thesis (Ph. D.)--Ohio State University, 2009. / Title from first page of PDF file. Includes bibliographical references (p. 40-42).

No description available.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.