Summary: We survey hyperinterpolation on the sphere \(\mathbb S^d\), \(d\geq 2\). The hyperinterpolation operator \(L_n\) is a linear projection onto the space \(P_n(\mathbb S^d)\) of spherical polynomials of degree \(\leq n\), which is obtained from \(L_2(\mathbb S^d)\)-orthogonal projection onto \(P_n(\mathbb S^d)\) 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 \(P_n(\mathbb S^d)\), which is relatively easy and inexpensive to compute. In contrast, the \(L_2(\mathbb S^d)\)-orthogonal projection onto \(P_n(\mathbb S^d)\) 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 \(P_n(\mathbb S^d)\) and with polynomial interpolation onto \(P_n(\mathbb S^d)\): 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, \(L_n:H^t(\mathbb S^d)\to H^s(\mathbb S^d)\), with \(t\geq s\geq 0\) and \(t > \frac d2\), 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.
For the entire collection see [Zbl 1098.00009].