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Characterisations of function spaces on fractals
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three papers, all of them on the topic of function spaces on fractals.

The papers summarised in this thesis are:

Paper I Mats Bodin, Wavelets and function spaces on Mauldin-Williams fractals, Research Report in Mathematics No. 7, Umeå University, 2005.

Paper II Mats Bodin, Harmonic functions and Lipschitz spaces on the Sierpinski gasket, Research Report in Mathematics No. 8, Umeå University, 2005.

Paper III Mats Bodin, A discrete characterisation of Lipschitz spaces on fractals, Manuscript.

The first paper deals with piecewise continuous wavelets of higher order in Besov spaces defined on fractals. A. Jonsson has constructed wavelets of higher order on fractals, and characterises Besov spaces on totally disconnected self-similar sets, by means of the magnitude of the coefficients in the wavelet expansion of the function. For a class of fractals, W. Jin shows that such wavelets can be constructed by recursively calculating moments. We extend their results to a class of graph directed self-similar fractals, introduced by R. D. Mauldin and S. C. Williams.

In the second paper we compare differently defined function spaces on the Sierpinski gasket. R. S. Strichartz proposes a discrete definition of Besov spaces of continuous functions on self-similar fractals having a regular harmonic structure. We identify some of them with Lipschitz spaces introduced by A. Jonsson, when the underlying domain is the Sierpinski gasket. We also characterise some of these spaces by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base.

The last paper gives a discrete characterisation of certain Lipschitz spaces on a class of fractal sets. A. Kamont has discretely characterised Besov spaces on intervals. We give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self-similar sets. This shows that, on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik , 2005. , 30 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 32
Keyword [en]
function spaces, wavelets, bases, fractals, triangulations, iterated function systems
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-580ISBN: 91-7305-932-3 (print)OAI: oai:DiVA.org:umu-580DiVA: diva2:143855
Public defence
2005-09-30
Available from: 2005-09-01 Created: 2005-09-01Bibliographically approved
List of papers
1. Wavelets and function spaces on Mauldin-Williams fractals
Open this publication in new window or tab >>Wavelets and function spaces on Mauldin-Williams fractals
2005 In: Research Report in Mathematics, Vol. 7Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-4657 (URN)
Available from: 2005-09-01 Created: 2005-09-01Bibliographically approved
2. Harmonic functions and Lipschitz spaces on the Sierpinski gasket
Open this publication in new window or tab >>Harmonic functions and Lipschitz spaces on the Sierpinski gasket
2005 In: Research Report in Mathematics, Vol. 8Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-4658 (URN)
Available from: 2005-09-01 Created: 2005-09-01Bibliographically approved
3. A discrete characterisation of Lipschitz spaces on fractals
Open this publication in new window or tab >>A discrete characterisation of Lipschitz spaces on fractals
2006 (English)Report (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå University, Department of Mathematics and Mathematical Statistics, 2006. 41 p.
Series
Research report in mathematics, ISSN 1653-0810 ; 6
Identifiers
urn:nbn:se:umu:diva-4659 (URN)
Available from: 2005-09-01 Created: 2005-09-01 Last updated: 2010-11-25Bibliographically approved

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