Optimality of the range for which equivalence between certain measures of smoothness holds
Volume 198 / 2010
Studia Mathematica 198 (2010), 271-277
MSC: 42B35, 41A17, 41A63.
DOI: 10.4064/sm198-3-6
Abstract
Recently it was proved for $1< p< \infty $ that $\omega ^m(f,t)_p,$ a modulus of smoothness on the unit sphere, and $\widetilde{K}_m(f,t^m)_p,$ a $K$-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range $1< p< \infty $ is optimal; that is, the equivalence $\omega ^m(f,t)_p\approx \widetilde{K}_m(f,t^r)_p$ does not hold either for $p=\infty $ or for $p=1.$
