Maximality and completeness of orthogonal exponentials on the cube
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| Format: | Preprint |
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2024
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| _version_ | 1866908284669132800 |
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| author | Kolountzakis, Mihail N. Lev, Nir Matolcsi, Máté |
| author_facet | Kolountzakis, Mihail N. Lev, Nir Matolcsi, Máté |
| contents | It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension $1$ there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_12387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximality and completeness of orthogonal exponentials on the cube Kolountzakis, Mihail N. Lev, Nir Matolcsi, Máté Classical Analysis and ODEs 42B10, 42C05, 52C22 It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension $1$ there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials. |
| title | Maximality and completeness of orthogonal exponentials on the cube |
| topic | Classical Analysis and ODEs 42B10, 42C05, 52C22 |
| url | https://arxiv.org/abs/2410.12387 |