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Cardinal spline wavelet decomposition based on quasi-interpolation and local projection
Thesis (MSc (Mathematics))--University of Stellenbosch, 2009.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : University of Stellenbosch
2009
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| _version_ | 1867614091833507840 |
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| access_status_str | Open Access |
| author | Ahiati, Veroncia Sitsofe |
| author2 | De Villiers, Johan |
| author_browse | Ahiati, Veroncia Sitsofe De Villiers, Johan |
| author_facet | De Villiers, Johan Ahiati, Veroncia Sitsofe |
| author_sort | Ahiati, Veroncia Sitsofe |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/2580 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:46:31.699Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2009 |
| publishDateRange | 2009 |
| publishDateSort | 2009 |
| 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:10019.1/2580 Cardinal spline wavelet decomposition based on quasi-interpolation and local projection Ahiati, Veroncia Sitsofe De Villiers, Johan University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Mathematics. Cardinal spline wavelet decomposition Quasi-interpolation Local projection Dissertations -- Mathematics Theses -- Mathematics Wavelets (Mathematics) Splines Thesis (MSc (Mathematics))--University of Stellenbosch, 2009. Wavelet decomposition techniques have grown over the last two decades into a powerful tool in signal analysis. Similarly, spline functions have enjoyed a sustained high popularity in the approximation of data. In this thesis, we study the cardinal B-spline wavelet construction procedure based on quasiinterpolation and local linear projection, before specialising to the cubic B-spline on a bounded interval. First, we present some fundamental results on cardinal B-splines, which are piecewise polynomials with uniformly spaced breakpoints at the dyadic points Z/2r, for r ∈ Z. We start our wavelet decomposition method with a quasi-interpolation operator Qm,r mapping, for every integer r, real-valued functions on R into Sr m where Sr m is the space of cardinal splines of order m, such that the polynomial reproduction property Qm,rp = p, p ∈ m−1, r ∈ Z is satisfied. We then give the explicit construction of Qm,r. We next introduce, in Chapter 3, a local linear projection operator sequence {Pm,r : r ∈ Z}, with Pm,r : Sr+1 m → Sr m , r ∈ Z, in terms of a Laurent polynomial m solution of minimally length which satisfies a certain Bezout identity based on the refinement mask symbol Am, which we give explicitly. With such a linear projection operator sequence, we define, in Chapter 4, the error space sequence Wr m = {f − Pm,rf : f ∈ Sr+1 m }. We then show by solving a certain Bezout identity that there exists a finitely supported function m ∈ S1 m such that, for every r ∈ Z, the integer shift sequence { m(2 · −j)} spans the linear space Wr m . According to our definition, we then call m the mth order cardinal B-spline wavelet. The wavelet decomposition algorithm based on the quasi-interpolation operator Qm,r, the local linear projection operator Pm,r, and the wavelet m, is then based on finite sequences, and is shown to possess, for a given signal f, the essential property of yielding relatively small wavelet coefficients in regions where the support interval of m(2r · −j) overlaps with a Cm-smooth region of f. Finally, in Chapter 5, we explicitly construct minimally supported cubic B-spline wavelets on a bounded interval [0, n]. We also develop a corresponding explicit decomposition algorithm for a signal f on a bounded interval. ii Throughout Chapters 2 to 5, numerical examples are provided to graphically illustrate the theoretical results. 2009-03-02T08:23:21Z 2010-06-01T08:52:48Z 2009-03-02T08:23:21Z 2010-06-01T08:52:48Z 2009-03 Thesis http://hdl.handle.net/10019.1/2580 en University of Stellenbosch application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Cardinal spline wavelet decomposition Quasi-interpolation Local projection Dissertations -- Mathematics Theses -- Mathematics Wavelets (Mathematics) Splines Ahiati, Veroncia Sitsofe Cardinal spline wavelet decomposition based on quasi-interpolation and local projection |
| title | Cardinal spline wavelet decomposition based on quasi-interpolation and local projection |
| title_full | Cardinal spline wavelet decomposition based on quasi-interpolation and local projection |
| title_fullStr | Cardinal spline wavelet decomposition based on quasi-interpolation and local projection |
| title_full_unstemmed | Cardinal spline wavelet decomposition based on quasi-interpolation and local projection |
| title_short | Cardinal spline wavelet decomposition based on quasi-interpolation and local projection |
| title_sort | cardinal spline wavelet decomposition based on quasi interpolation and local projection |
| topic | Cardinal spline wavelet decomposition Quasi-interpolation Local projection Dissertations -- Mathematics Theses -- Mathematics Wavelets (Mathematics) Splines |
| url | http://hdl.handle.net/10019.1/2580 |
| work_keys_str_mv | AT ahiativeronciasitsofe cardinalsplinewaveletdecompositionbasedonquasiinterpolationandlocalprojection |