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1
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Optimality of the range for which equivalence between certain measures of smoothness holds

100%
Ditzian Z.
Studia Mathematica
|
2010
|
tom 198
|
nr 3
271-277
EN
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.
2
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Moduli of smoothness of functions and their derivatives

63%
Ditzian Z., Tikhonov S.
Studia Mathematica
|
2007
|
tom 180
|
nr 2
143-160
EN
Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_{p}(T)$ and $L_{p}[-1,1]$ for 0 < p < ∞ using the moduli of smoothness $ω^{r}(f,t)_{p}$ and $ω^{r}_{φ}(f,t)_{p}$ respectively.
3
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Equivalence of measures of smoothness in $L_{p}(S^{d-1})$, 1 < p < ∞

51%
Dai F., Ditzian Z., Huang H.
Studia Mathematica
|
2010
|
tom 196
|
nr 2
179-205
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.
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