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Sharp Jackson-Stechkin inequality in \(L^2\) for multidimensional spheres. (English. Russian original) Zbl 0903.41014
Math. Notes 60, No. 3, 248-263 (1996); translation from Mat. Zametki 60, No. 3, 333-355 (1996).
For \(m\geq 2\), let \(\mathbb{S}^{m-1}=\{x\in\mathbb{R}^m:\| x\| =1\}\), where \(\| \cdot\|\) is the Euclidean norm and let \(L^2= L^2(\mathbb{S}^{m-1})\) be the associated Hilbert space of square integrable functions. For \(f\in L^2\) and for the set of polynomials \(P_n\) of degree at most \(n\), restricted to \(\mathbb{S}^{m-1}\), let \(E_n(f)\) be the best approximation of \(f\). The translation with step \(t > 0\) is the operator \(s_t:L^2\rightarrow L^2\) \[ s_tf(x)=\frac 1{| \mathbb{S}^{m-2}_x| }\int_{\mathbb{S}^{m-2}_x} f(x\cos t+\xi \sin t) d\xi, \] where \(\mathbb{S}^{m-2}_x=\{\xi\in \mathbb{S}^{m-1}: (x| \xi)=0\}\). Denoting by \(\psi_r(u)=(1-u)^{r/2}\), \(r>0\) and following Kh. P. Rustamov [Izv. Ross. Akad. Nauk, Ser. Math. 57, No. 5, 127-148 (1993; Zbl 0821.41016)] the operator \(\Delta^r_t=\sum_{k\geq 0} \frac 1{k!}\psi_r^{(k)}(0)s^k_t\) is called the \(r\)-th order difference operator with respect to the translation \(s_t\). Now the \(r\)-th order modulus of continuity of \(f\) is \[ \omega_r(f,\tau)=\sup\{\| \Delta^r_tf\| : 0 <t \leq\tau\}. \] The main results of the author are the following sharp Jackson-Stechkin inequality: Let \(n=1,2,\dots\). Then for \(m=2,3,\dots\) and \(\lambda=(m-2)/2\) each function \(f\in L^2\), \(f\neq \text{const.}\) satisfies \[ E_{n-1}(f)<\omega_r(f,2\tau_{n,\lambda}), \quad r\geq 1,\qquad E_{n-1}(f)<2^{(1-r)/2} \omega_r(f,2\tau_{n,\lambda}), \quad 0<r<1, \] where \(\tau_{n,\lambda}\) is the first positive zero of the Gegenbauer cosine polynomial \(C^\lambda_n(\cos t)\). A consistent history of the subject and an exhaustive bibliography are also included in the paper.

MSC:
41A50 Best approximation, Chebyshev systems
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