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Bergman space methods and integral means spectra of univalent functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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 [en]
Univalent functions, integral means spectra, Bergman spaces
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4354ISBN: 987-91-7178-630-2 OAI: oai:DiVA.org:kth-4354DiVA: diva2:11935
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
List of papers
1. Spectral notions for conformal maps: a survey
Open this publication in new window or tab >>Spectral notions for conformal maps: a survey
2008 (English)In: Computational methods in Function Theory, ISSN 1617-9447, Vol. 8, no 2, 447-474 p.Article in journal (Refereed) Published
Abstract [en]

The universal means spectrum of conformal mappingshas been studied extensively in recent years. In some situations,sharp results are available, in others, only upper and lower estimateshave been obtained so far. We review some of the classicalresults before discussing the recent work of Hedenmalm andShimorin on estimates of the universal means spectrum near theorigin. It is our ambition to explain how their method works andwhat its limitations are. We then move on to the recent studyof the universal means spectrum of bounded functions near thepoint two conducted by Baranov and Hedenmalm. A number ofopen problems related to these topics are pointed out together withsome auxilliary results which are interesting in their own right.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7044 (URN)
Note
QC 20010414Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2010-11-17Bibliographically approved
2. An estimate of the universal means spectrum of conformal mappings
Open this publication in new window or tab >>An estimate of the universal means spectrum of conformal mappings
2006 (English)In: Computational methods in Function Theory, ISSN 1617-9447, Vol. 6, no 2, 423-436 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7045 (URN)
Note
QC 20010414Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2010-11-17Bibliographically approved
3. Norm expansion along a zero variety
Open this publication in new window or tab >>Norm expansion along a zero variety
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).

Keyword
norm expansion; Bergman kernel expansion
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7046 (URN)10.1016/j.jfa.2007.09.011 (DOI)000254488800006 ()2-s2.0-39149131739 (Scopus ID)
Note

QC 20100414

Available from: 2007-05-04 Created: 2007-05-04 Last updated: 2017-06-14Bibliographically approved

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Citation style
  • apa
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  • sv-SE
  • Other locale
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Output format
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