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Norm expansion along a zero variety
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4971-7147
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2008 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 254, 1601-1625 p.Article in journal (Refereed) Published
Abstract [en]

The reproducing kernel function of a weighted Bergman space over domains in C-d is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d = 2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z(1) = z(2) as subvariety) and the ball (with z(2) = 0 as subvariety), as well as for a weighted Bargmann-Fock space over C-2 (with the diagonal z(1) = z(2) as subvariety).

Place, publisher, year, edition, pages
2008. Vol. 254, 1601-1625 p.
Keyword [en]
norm expansion; Bergman kernel expansion
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7046DOI: 10.1016/j.jfa.2007.09.011ISI: 000254488800006Scopus ID: 2-s2.0-39149131739OAI: oai:DiVA.org:kth-7046DiVA: diva2:11934
Note

QC 20100414

Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2017-06-14Bibliographically approved
In thesis
1. Bergman space methods and integral means spectra of univalent functions
Open this publication in new window or tab >>Bergman space methods and integral means spectra of univalent functions
2007 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

We study universal integral means spectra of certain classes of univalent functions defined on subsets of the complex plane. After reformulating the definition of the integral means spectrum of a univalent function in terms of membership in weighted Bergman spaces, we describe the Hilbert space techniques that can be used to estimate universal means spectra from above. Finally, we show that the method of norm expansion used in that context can be applied in a more general setting to reproducing kernel spaces in order to explicitly compute kernel functions.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. iv, 26 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 0704
Keyword
Univalent functions, integral means spectra, Bergman spaces
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4354 (URN)987-91-7178-630-2 (ISBN)
Presentation
Seminarierum 3721, KTH, Lindstedtvägen 25, Stockholm
Supervisors
Note
QC 20101117Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2010-11-17Bibliographically approved
2. Conformal Maps, Bergman Spaces, and Random Growth Models
Open this publication in new window or tab >>Conformal Maps, Bergman Spaces, and Random Growth Models
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and five research papers on topics related to conformal mapping, the Loewner equation and its applications, and Bergman-type spaces of holomorphic functions. The first two papers are devoted to the study of integral means of derivatives of conformal mappings. In Paper I, we present improved upper estimates of the universal means spectrum of conformal mappingsof the unit disk. These estimates rely on inequalities  obtained by Hedenmalm and Shimorin using Bergman space techniques, and on computer calculations. Paper II is a survey of recent results on the universal means spectrum, with particular emphasis on Bergman spacetechniques.Paper III concerns Bergman-type spaces of holomorphic functions in subsets of $\textbf{C}^d$ and their reproducing kernel functions. By expanding the norm of a function in a Bergman space along the zero variety of a polynomial, we obtain a series expansion of reproducing kernel functions in terms of kernels associated with lower-dimensionalspaces of holomorphic functions. We show how this general approach can be used to explicitly compute kernel functions for certain weighted Bergman and Bargmann-Fock spaces defined in domains in $\textbf{C}^2$.The last two papers contribute to the theory of Loewner chains and theirapplications in the analysis of planar random growth model defined in terms of compositions of conformal maps.In Paper IV, we study Loewner chains generated by unimodular L\'evy processes.We first establish the existence of a capacity scaling limit for the associated growing hulls in terms of whole-plane Loewner chains driven by a time-reversed process. We then analyze the properties of Loewner chains associated with a class of two-parameter compound Poisson processes, and we describe the dependence of the geometric properties of the hulls on the parameters of the driving process. In Paper V, we consider a variation of the Hastings-Levitov growth model, with anisotropic growth. We again establish results concerning scaling limits, when the number of compositions increases and the basic conformal mappings tends to the identity. We show that the resulting limit sets can be associated with solutions to the Loewner equation.We also prove that, in the limit, the evolution of harmonic measure on the boundary is deterministic and is determined by the flow associated with an ordinary differential equation, and we give a description of the fluctuations around this deterministic limit flow.

Place, publisher, year, edition, pages
Stockholm: KTH, 2010. vi, 46 p.
Series
TRITA-MAT. MA, ISSN 1401-2278 ; 10:03
Keyword
Conformal maps, Bergman kernels, planar growth models
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-12364 (URN)978-91-7415-593-8 (ISBN)
Public defence
2010-05-04, F3, Lindstedsv. 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20100414

Available from: 2010-04-14 Created: 2010-04-12 Last updated: 2017-02-23Bibliographically approved

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Publisher's full textScopushttp://dx.doi.org/10.1016/j.jfa.2007.09.01

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Hedenmalm, Håkan

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