Real and complex spherical designs and their Gramian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909896759312384 |
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| author | Waldron, Shayne |
| author_facet | Waldron, Shayne |
| contents | If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such spherical designs are determined by their Gramian (Gram matrix). We outline a general method to obtain such a characterisation as the minima of a function of the Gramian, which we call a potential. This characterisation can be used for the numerical and analytic construction of spherical designs. When the space P of polynomials is not irreducible under the action of the unitary group, then the potential is not unique. In several cases of interest, e.g., spherical t-designs and half-designs, we use this flexibility to provide potentials with a very simple form. We then use our results to develop certain aspects of the theory of real and complex spherical designs for unitarily invariant polynomial spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real and complex spherical designs and their Gramian Waldron, Shayne General Mathematics 05B25, 05B30, 42C15, 65D30 If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such spherical designs are determined by their Gramian (Gram matrix). We outline a general method to obtain such a characterisation as the minima of a function of the Gramian, which we call a potential. This characterisation can be used for the numerical and analytic construction of spherical designs. When the space P of polynomials is not irreducible under the action of the unitary group, then the potential is not unique. In several cases of interest, e.g., spherical t-designs and half-designs, we use this flexibility to provide potentials with a very simple form. We then use our results to develop certain aspects of the theory of real and complex spherical designs for unitarily invariant polynomial spaces. |
| title | Real and complex spherical designs and their Gramian |
| topic | General Mathematics 05B25, 05B30, 42C15, 65D30 |
| url | https://arxiv.org/abs/2511.07452 |