Equivalence of measures of smoothness in $L_{p}(S^{d-1})$, 1 < p < ∞

• Autorzy: F. Dai , Z. Ditzian , Hongwei Huang
• Języki publikacji: EN

Abstrakty

EN

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
$K̃ₘ(f,t^{m})_{p} ≡ inf{∥f - g∥_{L_{p}(S^{d-1})} + t^{m}∥(-Δ̃)^{m/2}g∥_{L_{p}(S^{d-1})}: g ∈ 𝓓((-Δ̃)^{m/2})}$
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.

Kategorie tematyczne

• 42B15: Multipliers
• 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 196
• Numer: 2
• Strony: 179-205

Daty

wydano

2010

Twórcy

autor

F. Dai

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

autor

Z. Ditzian

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

autor

Hongwei Huang

• School of Mathematical Sciences, Xiamen University, 361005, Xiamen, Fujian, China

Identyfikatory

DOI

10.4064/sm196-2-5