The Equivalent Theorem for Jackson-type Operators on Spherical Cap

This article discusses the approximation on the spherical cap. The Jackson-type operators {J(v,s)(m)}(v=11)(infinity) defined on the spherical cap D(x(0), gamma) are used to approximate the spherical function. An equivalent theorem, which simultaneously contains the direct and inverse inequalities for the approximation, is established by means of the modulus of smoothness on the spherical cap. Namely, for any p-th (1 <= p < infinity) integrable or continuous function f defined on the spherical cap D(x(0), gamma), it is proved that there exist positive constants C-1 and C-2, such that C-1 omega(2) (f, 1/v)(D.p) <=parallel to J(v,s)(m) (f) - f parallel to D,p <= C-2 omega(2) (f, 1/v)(D,p), where omega(2) (f, t)(D,p) is the modulus of smoothness of degree 2 of f on the spherical cap D(x(0), gamma).