The algebraic structure of hyperinterpolation class on the sphere

This paper investigates the algebraic properties of the hyperinterpolation class $\mathbf{H}\mathbf{C}({\mathbb{S}}^d)$ 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.