Moving finite unit tight frames for \(S^n\)
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| Publicat a: | arXiv.org (Sep 25, 2012), p. n/a |
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| Autor principal: | |
| Altres autors: | , |
| Publicat: |
Cornell University Library, arXiv.org
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| Matèries: | |
| Accés en línia: | Citation/Abstract Full text outside of ProQuest |
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| Resum: | Frames for \(\R^n\) can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are \(S^1\), \(S^3\), and \(S^7\). On the other hand, after combining the two separate meanings of the word "frame", we show that the \(n\)-dimensional sphere, \(S^n\), has a moving finite unit tight frame for its tangent bundle if and only if \(n\) is odd. We give a procedure for creating vector fields on \(S^{2n-1}\) for all \(n\in\N\), and we characterize exactly when sets of such vector fields form a moving finite unit tight frame. |
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| ISSN: | 2331-8422 |
| Font: | Engineering Database |