球面上超插值类的代数结构
The algebraic structure of hyperinterpolation class on the sphere
AI总结：
本文研究单位球面上超插值类的代数性质，基于离散（半）内积框架构建相关算子与超半群理论，分析四类超插值算子的代数特性并证明多项相关结论。
摘要：
本文研究单位球面$\mathbb{S}^d$上超插值类$\mathbf{HC}(\mathbb{S}^d)$的代数性质，聚焦由具有有界$L_2$算子范数的经典超插值导出的算子。借助离散（半）内积框架，我们构建了超自伴算子、超投影算子和超半群的理论，分析了滤波超插值、Lasso超插值、硬阈值超插值和广义超插值四类具体算子。我们证明广义超插值算子是超自伴的，且与超插值算子可交换；此外，硬阈值与经典超插值算子构成超半群，其中硬阈值超插值是极小素超理想。最后，我们证实超插值算子在超半群上充当超同态。
英文摘要：
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.
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