Wavelet characterizations of the Sobolev wavefront set: bandlimited wavelets and compactly supported wavelets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911770950500352 |
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| author | Führ, Hartmut Ghandehari, Mahya |
| author_facet | Führ, Hartmut Ghandehari, Mahya |
| contents | We consider the problem of characterizing the Sobolev wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, with the latter being defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. We derive necessary and sufficient criteria for elements of the Sobolev wavefront set, formulated in terms of the decay behaviour of a given generalized continuous wavelet transform. It turns out that the characterization of directed smoothness of finite order can be performed in the two important cases: (1) bandlimited wavelets, and (2) wavelets with finitely many vanishing moments (e.g.~compactly supported wavelets).
The main results of this paper are based on a number of fairly technical conditions on the dilation group. In order to demonstrate their wide applicability, we exhibit a large class of generalized shearlet groups in arbitrary dimensions fulfilling all required conditions, and give estimates of the involved constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wavelet characterizations of the Sobolev wavefront set: bandlimited wavelets and compactly supported wavelets Führ, Hartmut Ghandehari, Mahya Functional Analysis 42C15, 42C40, 46F12 We consider the problem of characterizing the Sobolev wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, with the latter being defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. We derive necessary and sufficient criteria for elements of the Sobolev wavefront set, formulated in terms of the decay behaviour of a given generalized continuous wavelet transform. It turns out that the characterization of directed smoothness of finite order can be performed in the two important cases: (1) bandlimited wavelets, and (2) wavelets with finitely many vanishing moments (e.g.~compactly supported wavelets). The main results of this paper are based on a number of fairly technical conditions on the dilation group. In order to demonstrate their wide applicability, we exhibit a large class of generalized shearlet groups in arbitrary dimensions fulfilling all required conditions, and give estimates of the involved constants. |
| title | Wavelet characterizations of the Sobolev wavefront set: bandlimited wavelets and compactly supported wavelets |
| topic | Functional Analysis 42C15, 42C40, 46F12 |
| url | https://arxiv.org/abs/2402.02796 |