Hyperinterpolation on the sphere

In this paper we survey hyperinterpolation on the sphere $\\mathbb{S}^d$, $d\\geq 2$. The hyperinterpolation operator $L_n$ is a linear projection onto the space $\\mathbb{P}_n(\\mathbb{S}^d)$ of spherical polynomials of degree $\\leq n$, which is obtained from $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ by discretizing the integrals in the $L_2(\\mathbb{S}^d)$ inner products by a positive-weight numerical integration rule of polynomial degree of exactness $2n$. Thus hyperinterpolation is a kind of `discretized orthogonal projection' onto $\\mathbb{P}_n(\\mathbb{S}^d)$, which is relatively easy and inexpensive to compute. In contrast, the $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ cannot generally be computed without some discretization of the integrals in the inner products; hyperinterpolation is a realization of such a discretization. We compare hyperinterpolation with $L_2(\\mathbb{S}^d)$-orthogonal projection onto $\\mathbb{P}_n(\\mathbb{S}^d)$ and with polynomial interpolation onto $\\mathbb{P}_n(\\mathbb{S}^d)$: we discuss the properties, estimates of the operator norms in terms of $n$, and estimates of the approximation error. We also present a new estimate of the approximation error of hyperinterpolation in the Sobolev space setting, that is, $L_n:H^t(\\mathbb{S}^d)\\rightarrow H^s(\\mathbb{S}^d)$, with $t\\geq s\\geq 0$ and $t>d/2$, where $H^s(\\mathbb{S}^d)$ is for integer $s$ roughly the Sobolev space of those functions whose generalized derivatives up to the order $s$ are square-integrable.