The direct and converse inequalities for jackson-type operators on spherical cap

Abstract

Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, the so-called Jackson-type operator { J k,sm } k=1∞ is constructed to approximate the function defined on the spherical cap D(x 0, γ). We thus establish the direct and inverse inequalities and obtain saturation theorems for { J k,sm } k=1∞ on the cap D(x 0, γ). Using methods of K -functional and multiplier, we obtain the inequality C1 ∥J k,sm (f)-f ∥ D,p ≤ ω2 (f, k -1) D,p ≤ C 2 max v≥k ∥ J v,sm (f)-f ∥ D,p and that the saturation order of these operators is O(k -2), where ω2 (f,t) D,p is the modulus of smoothness of degree 2, the constants C 1 and C 2 are independent of k and f. Copyright © 2009 Y. Wang and F. Cao.