Quadrature and Sharp Embeddings in Besov Spaces on the Sphere with Exponential Weighted Polynomials

Abstract

We study mainly the worst-case in a Sobolev space setting for cubature and extremal systems of points with numerical integration over the sphere. An approximation of the constructive polynomial on the sphere was considered. We Find the optimal lower bounds for cubature error in Sobolev spaces of arbitrary order. The quadrature in Besov spaces on the Euclidion sphere was shown. We obtain the sobolev error estimates and determined the Bernstein inequality for scattered data interpolation, we also establish the spherical basis functions and construct the uniform distribution of points on spheres .we gine the structure of the orthogonal, inequalities and orthonormal polynomials with exponential-type weights . we investigate the sharp embeddings of Besov-type spaces involving only logarithmic smoothness.