A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets
By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the k-space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic function...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2012/758694 |
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| author | Felipe A. Apolonio Daniel H. T. Franco Fábio N. Fagundes |
| author_facet | Felipe A. Apolonio Daniel H. T. Franco Fábio N. Fagundes |
| author_sort | Felipe A. Apolonio |
| collection | DOAJ |
| description | By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the k-space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distribution f∈𝒮′(ℝn), the continuous wavelet transform of f with respect to a conical wavelet is defined in such a way that the directional wavelet transform of f yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of f. |
| format | Article |
| id | doaj-art-2b23b92febb24a3182da9cf074284b32 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-2b23b92febb24a3182da9cf074284b322025-02-03T06:13:24ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252012-01-01201210.1155/2012/758694758694A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront SetsFelipe A. Apolonio0Daniel H. T. Franco1Fábio N. Fagundes2Departamento de Física, Universidade Federal de Viçosa, Avenida Peter Henry Rolfs s/n, Campus Universitário Viçosa, 36570-000 Vioçsa, MG, BrazilDepartamento de Física, Universidade Federal de Viçosa, Avenida Peter Henry Rolfs s/n, Campus Universitário Viçosa, 36570-000 Vioçsa, MG, BrazilUniversidade Federal de Mato Grosso, Campus Sinop, ICNHS, Avenida Alexandre Ferronato 1.200, Distrito Industrial, 78557-267 Sinop, MT, BrazilBy using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the k-space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distribution f∈𝒮′(ℝn), the continuous wavelet transform of f with respect to a conical wavelet is defined in such a way that the directional wavelet transform of f yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of f.http://dx.doi.org/10.1155/2012/758694 |
| spellingShingle | Felipe A. Apolonio Daniel H. T. Franco Fábio N. Fagundes A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets International Journal of Mathematics and Mathematical Sciences |
| title | A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets |
| title_full | A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets |
| title_fullStr | A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets |
| title_full_unstemmed | A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets |
| title_short | A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets |
| title_sort | note on directional wavelet transform distributional boundary values and analytic wavefront sets |
| url | http://dx.doi.org/10.1155/2012/758694 |
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