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Multiwavelet analysis on fractals
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and a summary, followed by two papers, both of them on the topic of function spaces on fractals.

Paper I: Andreas Brodin, Pointwise Convergence of Haar type Wavelets on Self-Similar Sets, Manuscript.

Paper II: Andreas Brodin,

Regularization of Wavelet Expansion characterizes Besov Spaces on Fractals, Manuscript.

Properties of wavelets, originally constructed by A. Jonsson, are studied in both papers. The wavelets are piecewise polynomial functions on self-similar fractal sets.

In Paper I, pointwise convergence of partial sums of the wavelet expansion is investigated. On a specific fractal set, the Sierpinski gasket, pointwise convergence of the partial sums is shown by calculating the kernel explicitly, when the wavelets are piecewise constant functions. For more general self-similar fractals, pointwise convergence of the partial sums and their derivatives, in case the expanded function has higher regularity, is shown using a different technique based on Markov's inequality.

A. Jonsson has shown that on a class of totally disconnected self-similar sets it is possible to characterize Besov spaces by means of the magnitude of the coefficients in the wavelet expansion of a function. M. Bodin has extended his results to a class of graph directed self-similar sets introduced by Mauldin and Williams. Unfortunately, these results only holds for fractals such that the sets in the first generation of the fractal are disjoint. In Paper II we are able to characterize Besov spaces on a class of fractals not necessarily sharing this condition by making the wavelet expansion smooth. We create continuous regularizations of the partial sums of the wavelet expansion and show that properties of these regularizations can be used to characterize Besov spaces.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik , 2007. , 19 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 38
Keyword [en]
Wavelets, Fractals, Pointwise convergence, Besov spaces, Regularization
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-1131ISBN: 91-7264-334-5 OAI: oai:DiVA.org:umu-1131DiVA: diva2:140319
Public defence
2007-06-07, MA121, MIT-huset, SE-901 87, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2007-05-10 Created: 2007-05-10 Last updated: 2009-09-07Bibliographically approved
List of papers
1. Pointwise convergence of Haar type wavelets on self-similar sets
Open this publication in new window or tab >>Pointwise convergence of Haar type wavelets on self-similar sets
2006 (English)Report (Other academic)
Series
Research report in mathematics, ISSN 1653-0810 ; 5
Identifiers
urn:nbn:se:umu:diva-8212 (URN)
Available from: 2008-01-15 Created: 2008-01-15 Last updated: 2009-09-07Bibliographically approved
2. Regularization of wavelet expansion characterizes Besov spaces on fractals
Open this publication in new window or tab >>Regularization of wavelet expansion characterizes Besov spaces on fractals
Manuscript (Other academic)
Identifiers
urn:nbn:se:umu:diva-2360 (URN)
Available from: 2007-05-10 Created: 2007-05-10 Last updated: 2010-01-13Bibliographically approved

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CiteExportLink to record
Permanent link

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Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf