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On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence
Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : University of Stellenbosch
2008
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| _version_ | 1867613914523500544 |
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| access_status_str | Open Access |
| author | De Wet, Wouter de Vos |
| author2 | De Villiers, J. M. |
| author_browse | De Villiers, J. M. De Wet, Wouter de Vos |
| author_facet | De Villiers, J. M. De Wet, Wouter de Vos |
| author_sort | De Wet, Wouter de Vos |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/1480 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:43:43.080Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2008 |
| publishDateRange | 2008 |
| publishDateSort | 2008 |
| publisher | Stellenbosch : University of Stellenbosch |
| publisherStr | Stellenbosch : University of Stellenbosch |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/1480 On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence De Wet, Wouter de Vos De Villiers, J. M. University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Refinable function Mask factorisation Regularity Subdivision Convergence Factorization (Mathematics) Dissertations -- Mathematics Theses -- Mathematics Mathematical Sciences Mathematics Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. We study refinable functions where the dilation factor is not always assumed to be 2. In our investigation, the role of convolutions and refinable step functions is emphasized as a framework for understanding various previously published results. Of particular importance is a class of polynomial factors, which was first introduced for dilation factor 2 by Berg and Plonka and which we generalise to general integer dilation factors. We obtain results on the existence of refinable functions corresponding to certain reduced masks which generalise similar results for dilation factor 2, where our proofs do not rely on Fourier methods as those in the existing literature do. We also consider subdivision for general integer dilation factors. In this regard, we extend previous results of De Villiers on refinable function existence and subdivision convergence in the case of positive masks from dilation factor 2 to general integer dilation factors. We also obtain results on the preservation of subdivision convergence, as well as on the convergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomial factors are added to the mask symbol. We obtain sufficient conditions for the occurrence of polynomial sections in refinable functions and construct families of related refinable functions. We also obtain results on the regularity of a refinable function in terms of the mask symbol factorisation. In this regard, we obtain much more general sufficient conditions than those previously published, while for dilation factor 2, we obtain a characterisation of refinable functions with a given number of continuous derivatives. We also study the phenomenon of subsequence convergence in subdivision, which explains some of the behaviour that we observed in non-convergent subdivision processes during numerical experimentation. Here we are able to establish different sets of sufficient conditions for this to occur, with some results similar to standard subdivision convergence, e.g. that the limit function is refinable. These results provide generalisations of the corresponding results for subdivision, since subsequence convergence is a generalisation of subdivision convergence. The nature of this phenomenon is such that the standard subdivision algorithm can be extended in a trivial manner to allow it to work in instances where it previously failed. Lastly, we show how, for masks of length 3, explicit formulas for refinable functions can be used to calculate the exact values of the refinable function at rational points. Various examples with accompanying figures are given throughout the text to illustrate our results. Doctoral 2008-02-19T08:09:55Z 2010-06-01T08:22:46Z 2008-02-19T08:09:55Z 2010-06-01T08:22:46Z 2007-12 Thesis http://hdl.handle.net/10019.1/1480 en University of Stellenbosch 2103655 bytes application/pdf application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Refinable function Mask factorisation Regularity Subdivision Convergence Factorization (Mathematics) Dissertations -- Mathematics Theses -- Mathematics Mathematical Sciences Mathematics De Wet, Wouter de Vos On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence |
| title | On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence |
| title_full | On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence |
| title_fullStr | On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence |
| title_full_unstemmed | On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence |
| title_short | On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence |
| title_sort | on the analysis of refinable functions with respect to mask factorisation regularity and corresponding subdivision convergence |
| topic | Refinable function Mask factorisation Regularity Subdivision Convergence Factorization (Mathematics) Dissertations -- Mathematics Theses -- Mathematics Mathematical Sciences Mathematics |
| url | http://hdl.handle.net/10019.1/1480 |
| work_keys_str_mv | AT dewetwouterdevos ontheanalysisofrefinablefunctionswithrespecttomaskfactorisationregularityandcorrespondingsubdivisionconvergence |