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On the regularity of refinable functions
Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : University of Stellenbosch
2006
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| _version_ | 1867614137593364480 |
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| access_status_str | Open Access |
| author | Onwunta, Akwum A. |
| author2 | De Villiers, J. M. |
| author_browse | De Villiers, J. M. Onwunta, Akwum A. |
| author_facet | De Villiers, J. M. Onwunta, Akwum A. |
| author_sort | Onwunta, Akwum A. |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/2881 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:47:15.645Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2006 |
| publishDateRange | 2006 |
| publishDateSort | 2006 |
| 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:10019.1/2881 On the regularity of refinable functions Onwunta, Akwum A. De Villiers, J. M. University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Dissertations -- Mathematics Theses -- Mathematics Functions Function spaces Fourier analysis Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006. This work studies the regularity (or smoothness) of continuous finitely supported refinable functions which are mainly encountered in multiresolution analysis, iterative interpolation processes, signal analysis, etc. Here, we present various kinds of sufficient conditions on a given mask to guarantee the regularity class of the corresponding refinable function. First, we introduce and analyze the cardinal B-splines Nm, m ∈ N. In particular, we show that these functions are refinable and belong to the smoothness class Cm−2(R). As a generalization of the cardinal B-splines, we proceed to discuss refinable functions with positive mask coefficients. A standard result on the existence of a refinable function in the case of positive masks is quoted. Following [13], we extend the regularity result in [25], and we provide an example which illustrates the fact that the associated symbol to a given positive mask need not be a Hurwitz polynomial for its corresponding refinable function to be in a specified smoothness class. Furthermore, we apply our regularity result to an integral equation. An important tool for our work is Fourier analysis, from which we state some standard results and give the proof of a non-standard result. Next, we study the H¨older regularity of refinable functions, whose associated mask coefficients are not necessarily positive, by estimating the rate of decay of their Fourier transforms. After showing the embedding of certain Sobolev spaces into a H¨older regularity space, we proceed to discuss sufficient conditions for a given refinable function to be in such a H¨older space. We specifically express the minimum H¨older regularity of refinable functions as a function of the spectral radius of an associated transfer operator acting on a finite dimensional space of trigonometric polynomials. We apply our Fourier-based regularity results to the Daubechies and Dubuc-Deslauriers refinable functions, as well as to a one-parameter family of refinable functions, and then compare our regularity estimates with those obtained by means of a subdivision-based result from [28]. Moreover, we provide graphical examples to illustrate the theory developed. 2006-10-13T09:01:00Z 2010-06-01T09:00:45Z 2006-10-13T09:01:00Z 2010-06-01T09:00:45Z 2006-03 Thesis http://hdl.handle.net/10019.1/2881 en University of Stellenbosch 744842 bytes application/pdf application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Dissertations -- Mathematics Theses -- Mathematics Functions Function spaces Fourier analysis Onwunta, Akwum A. On the regularity of refinable functions |
| title | On the regularity of refinable functions |
| title_full | On the regularity of refinable functions |
| title_fullStr | On the regularity of refinable functions |
| title_full_unstemmed | On the regularity of refinable functions |
| title_short | On the regularity of refinable functions |
| title_sort | on the regularity of refinable functions |
| topic | Dissertations -- Mathematics Theses -- Mathematics Functions Function spaces Fourier analysis |
| url | http://hdl.handle.net/10019.1/2881 |
| work_keys_str_mv | AT onwuntaakwuma ontheregularityofrefinablefunctions |