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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
Full Text:
##### References:
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