Equivalence of measures of smoothness in $L_p\left(S^{d-1}\right)$, 1 < p < ∞

F. Dai, Z. Ditzian, and Hongwei Huang. "Equivalence of measures of smoothness in $L_{p}(S^{d-1})$, 1 < p < ∞." Studia Mathematica 196.2 (2010): 179-205. <http://eudml.org/doc/285537>.

@article{F2010,
abstract = {Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^\{d-1\}, Δ^\{k\}_\{ρ\}f(x) = Δ_\{ρ\}Δ^\{k-1\}_\{ρ\}f(x)$ and $Δ_\{ρ\}f(x) = f(ρx) - f(x)$ where ρ ∈ SO(d). Then $ω^\{m\}(f,t)_\{L_\{p\}(S^\{d-1\})\} ≡ sup\{∥Δ^\{m\}_\{ρ\}f∥_\{L_\{p\}(S^\{d-1\})\}: ρ ∈ SO(d), max_\{x∈ S^\{d-1\}\} ρx·x ≥ cos t\}$ and $K̃ₘ(f,t^\{m\})_\{p\} ≡ inf\{∥f - g∥_\{L_\{p\}(S^\{d-1\})\} + t^\{m\}∥(-Δ̃)^\{m/2\}g∥_\{L_\{p\}(S^\{d-1\})\}: g ∈ ((-Δ̃)^\{m/2\})\}$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_\{p\}(S^\{d-1\})$ given in this paper plays a significant role in the proof.},
author = {F. Dai, Z. Ditzian, Hongwei Huang},
journal = {Studia Mathematica},
keywords = {modulus of smoothness; -functional; Laplace-Beltrami operator on the sphere; sharp Jackson inequality},
language = {eng},
number = {2},
pages = {179-205},
title = {Equivalence of measures of smoothness in $L_\{p\}(S^\{d-1\})$, 1 < p < ∞},
url = {http://eudml.org/doc/285537},
volume = {196},
year = {2010},
}

TY - JOUR
AU - F. Dai
AU - Z. Ditzian
AU - Hongwei Huang
TI - Equivalence of measures of smoothness in $L_{p}(S^{d-1})$, 1 < p < ∞
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 2
SP - 179
EP - 205
AB - Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d-1}, Δ^{k}_{ρ}f(x) = Δ_{ρ}Δ^{k-1}_{ρ}f(x)$ and $Δ_{ρ}f(x) = f(ρx) - f(x)$ where ρ ∈ SO(d). Then $ω^{m}(f,t)_{L_{p}(S^{d-1})} ≡ sup{∥Δ^{m}_{ρ}f∥_{L_{p}(S^{d-1})}: ρ ∈ SO(d), max_{x∈ S^{d-1}} ρx·x ≥ cos t}$ and $K̃ₘ(f,t^{m})_{p} ≡ inf{∥f - g∥_{L_{p}(S^{d-1})} + t^{m}∥(-Δ̃)^{m/2}g∥_{L_{p}(S^{d-1})}: g ∈ ((-Δ̃)^{m/2})}$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_{p}(S^{d-1})$ given in this paper plays a significant role in the proof.
LA - eng
KW - modulus of smoothness; -functional; Laplace-Beltrami operator on the sphere; sharp Jackson inequality
UR - http://eudml.org/doc/285537
ER -