Optimal Lower Estimates for the Worst Case Cubature Error and the Approximation by Hyperinterpolation Operators in the Sobolev Space Setting on the Sphere

In this paper, we study two problems: one is of the worst case cubature error over the Sobolev classes on the sphere and another is of the approximation by hyperinterpolation operators on the sphere in the Sobolev space setting. We obtain lower estimates for the worst case cubature error over the Sobolev classes, which are optimal in the sense of order. We also obtain optimal estimates for the approximation by hyperinterpolation operators on the sphere in the Sobolev space setting.