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Abstrakty
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.
$ω^{m}(f,t)_{L_{p}(S^{d-1})} ≡ sup{∥Δ^{m}_{ρ}f∥_{L_{p}(S^{d-1})}: ρ ∈ SO(d), max_{x∈ S^{d-1}} ρx·x ≥ cos t}$
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
179-205
Opis fizyczny
Daty
wydano
2010
Twórcy
autor
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
autor
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
autor
- School of Mathematical Sciences, Xiamen University, 361005, Xiamen, Fujian, China
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-2-5