Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere.Thus cubature formula plays an important role in computing these spherical integrals.This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks.The authors give an existence proof of the exact positive cubature formula for spherical basis function networks,and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.
Bernstein inequality played an important role in approximation theory and Fourier analysis.This article first introduces a general system of functions and the socalled multivariate weighted Bernstein,Nikol'ski,and Ul'yanov-type inequalities.Then,the relations among these three inequalities are discussed.Namely,it is proved that a family of functions equipped with Bernstein-type inequality satisfies Nikol'ski-type and Ul'yanov-type inequality.Finally,as applications,some classical inequalities are deduced from the obtained results.
