Rate of approximation of combinations of averages on the spheres is shown to be equivalent to K-functionals yielding higher degree of smoothness. Results relating combinations of averages on rims of caps of spheres are also achieved.
About twenty years ago the measure of smoothness ω r ϕ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time.
For Lp spaces on T d , R d and S d−1 sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on T d , R d and S d−1) for which M (u 1/q) is convex for some q, 1 < q ≤ 2, where M (u) is the Orlicz function. Sharp converse inequalities for such spaces are deduced.