Optimality of the range for which equivalence between certain measures of smoothness holds

Z. Ditzian¹

¹ Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB, Canada T6G 2G1

Studia Mathematica, Tome 198 (2010) no. 3, pp. 271-277

Résumé

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 $p=1.$

Détail

DOI : 10.4064/sm198-3-6

Keywords: recently proved infty omega modulus smoothness unit sphere widetilde m k functional involving laplace beltrami operator equivalent shown range infty optimal equivalence omega approx widetilde r does either infty