The algebraic structure of hyperinterpolation class on the sphere
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911087287336960 |
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| author | An, Congpei Ran, Jiashu |
| author_facet | An, Congpei Ran, Jiashu |
| contents | This paper investigates the algebraic properties of the hyperinterpolation class $\mathbf{HC}(\mathbb{S}^d)$ on the unit sphere $ \mathbb{S}^d $. We focus on operators derived from the classical hyperinterpolation with bounded $ L_2 $ operator norms. By utilizing a discrete (semi) inner product framework, we develop the theory of hyper self-adjoint operators, hyper projection operators, and hyper semigroups. We analyze four specific operators: filtered, Lasso, hard thresholding, and generalized hyperinterpolations. We prove that the generalized hyperinterpolation operator is hyper self-adjoint and commutative with the hyperinterpolation operator. Additionally, we demonstrate that hard thresholding and classical hyperinterpolation operators form a hyper semigroup, with hard thresholding hyperinterpolation constituting the minimal prime hyper ideal. Finally, we establish that hyperinterpolation operators act as hyper homomorphisms on the hyper semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The algebraic structure of hyperinterpolation class on the sphere An, Congpei Ran, Jiashu Functional Analysis Numerical Analysis 41A10, 41A36, 47L20, 47L80 This paper investigates the algebraic properties of the hyperinterpolation class $\mathbf{HC}(\mathbb{S}^d)$ on the unit sphere $ \mathbb{S}^d $. We focus on operators derived from the classical hyperinterpolation with bounded $ L_2 $ operator norms. By utilizing a discrete (semi) inner product framework, we develop the theory of hyper self-adjoint operators, hyper projection operators, and hyper semigroups. We analyze four specific operators: filtered, Lasso, hard thresholding, and generalized hyperinterpolations. We prove that the generalized hyperinterpolation operator is hyper self-adjoint and commutative with the hyperinterpolation operator. Additionally, we demonstrate that hard thresholding and classical hyperinterpolation operators form a hyper semigroup, with hard thresholding hyperinterpolation constituting the minimal prime hyper ideal. Finally, we establish that hyperinterpolation operators act as hyper homomorphisms on the hyper semigroup. |
| title | The algebraic structure of hyperinterpolation class on the sphere |
| topic | Functional Analysis Numerical Analysis 41A10, 41A36, 47L20, 47L80 |
| url | https://arxiv.org/abs/2404.00523 |