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

Z. Ditzian. "Optimality of the range for which equivalence between certain measures of smoothness holds." Studia Mathematica 198.3 (2010): 271-277. <http://eudml.org/doc/285743>.

@article{Z2010,
abstract = {Recently it was proved for 1 < p < ∞ that $ω^\{m\}(f,t)_\{p\}$, a modulus of smoothness on the unit sphere, and $K̃ₘ(f,t^\{m\})_\{p\}$, a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence $ω^\{m\}(f,t)_\{p\} ≈ K̃ₘ(f,t^\{r\})_\{p\}$ does not hold either for p = ∞ or for p = 1.},
author = {Z. Ditzian},
journal = {Studia Mathematica},
language = {eng},
number = {3},
pages = {271-277},
title = {Optimality of the range for which equivalence between certain measures of smoothness holds},
url = {http://eudml.org/doc/285743},
volume = {198},
year = {2010},
}

TY - JOUR
AU - Z. Ditzian
TI - Optimality of the range for which equivalence between certain measures of smoothness holds
JO - Studia Mathematica
PY - 2010
VL - 198
IS - 3
SP - 271
EP - 277
AB - Recently it was proved for 1 < p < ∞ that $ω^{m}(f,t)_{p}$, a modulus of smoothness on the unit sphere, and $K̃ₘ(f,t^{m})_{p}$, a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence $ω^{m}(f,t)_{p} ≈ K̃ₘ(f,t^{r})_{p}$ does not hold either for p = ∞ or for p = 1.
LA - eng
UR - http://eudml.org/doc/285743
ER -