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The sharp Jackson inequality in the space \(L_p\) on the sphere. (English. Russian original) Zbl 1058.41502
Math. Notes 66, No. 1, 40-50 (1999); translation from Mat. Zametki 66, No. 1, 50-62 (1999).
The author gives a proof of his theorem which was announced in [”The Jackson theorem in \( L_p(S^{n-1})\)” (Russian), in Voronezh Winter School on Modern Methods in the Theory of Functions and Related Problems in Applied Mathematics and Mechanics (abstracts), p. 56, Izdat. Voronezh. Gos. Univ., Voronezh, 1997; per bibl.].
Let \(S^{n-1}=\{x\in\mathbb R^n\colon |x|=1\}, \Gamma_{\alpha}(x)=\{y\in S^{n-1}\colon xy=\cos\alpha\}\), and \(\sigma\) and \(\gamma_{\alpha}\) be normalized measures on \(S^{n-1}\) and \(\Gamma_{\alpha}(x)\) respectively. Let \[ \begin{split} L_p(S^{n-1})=\Big\{f\colon S^{n-1}\to \mathbb C| \| f\|_p=\Big(\int_{S^{n-1}} |f|^p \, d\sigma\Big)^{1/p}<\infty\Big\},\\ 1\leq p\leq\infty,\end{split} \] and let \[ E_R(f,S^{n-1})_p=\inf\Big\{\|f-g\|_p\colon g\in\sum_{l=0}^{R-1}\bigoplus H_l\Big\},\quad R\in\mathbb N, \] be the best approximation of the function \(f\in L_p(S^{n-1})\) by spherical harmonics \(H_l\) of degrees \(\leq R-1\). Let \[ \omega(\delta,f,S^{n-1})_p=\sup_{\alpha\leq\delta}\Big(\int_{S^{n-1}} \int_{\Gamma_{\alpha}(x)}|f(y)-f(x)|^p\, d_{\gamma_{\alpha}}(y)\, d\sigma(x) \Big)^{1/p}, \] \[ D(\delta,R,S^{n-1})_p=\sup_{f\in L_p(S^{n-1})}\frac{E_R(f,S^{n-1})_p} {\omega(\delta,f,S^{n-1})_p} \] be the exact constant in the Jackson inequality and \(t_R=\cos\tau_R\) be the greatest zero of Gegenbauer’s polynomial \(P_R(t)\).
Theorem. If \(n\geq 3, 1\leq p<2\), then \[ D(2\tau_R,2R-1,S^{n-1})_p=2^{-1/p'},\quad p'=p/(p-1). \]

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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