Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?

Singular and oscillatory functions play a crucial role in various applications, and their approximation is crucial for solving applied mathematics problems efficiently. Hyperinterpolation is a discrete projection method approximating functions with the $L^2$ orthogonal projection coefficients obtained by numerical integration. However, this approach may be inefficient for approximating singular and oscillatory functions, requiring a large number of integration points to achieve satisfactory accuracy. To address this issue, we propose a new approximation scheme in this paper, called efficient hyperinterpolation, which leverages the product-integration methods to attain the desired accuracy with fewer numerical integration points than the original scheme. We provide theorems that explain the superiority of efficient hyperinterpolation over the original scheme in approximating such functions belonging to $L^1$, $L^2$, and continuous function spaces, respectively, and demonstrate through numerical experiments on the interval and the sphere that our approach outperforms the original method in terms of accuracy when using a limited number of integration points.