The work was published in: Mat. Zametki 60 (1996), no. 3, pp. 333-355;
English translation: Math. Notes. 60 (1996), no. 3, pp. 248-263.

S U M M A R Y

41A17 41A44

B a b e n k o A. G.
Exact Jackson-Stechkin type inequality in $L^2$ space of functions on the multidimensional sphere.

In this paper we prove Jackson-Stechkin type inequality

$$E_{n-1}(f) < \omega_r (f,2 \tau _{n,\lambda }),\quad n\geq 1,... ...,\quad f\in L^2({\bf S}^{m-1}),\quad f\not \equiv \mbox{const},$$

which is exact for every $n=2,3,\ldots;$ here $E_{n-1}(f)$ is the best approximation of the function $f$ by spherical polynomials of degree $\leq n-1,\ \omega _r(f,\tau)$ is the $r$th modulus of continuity of $f$ based on translation

$$s_t f(x)=\frac{1}{\vert{\bf S}^{m-2}\vert}\int_{{\bf S}^{m-2}} f(x\cos t+\xi\sin t)d\xi ,$$

$$t\in{\bf R},\quad x\in{\bf S}^{m-1},\quad {\bf S}^{m-2}={\bf S}^{m-2}_x= \{\xi\in{\bf S}^{m-1} : x\cdot\xi=0\},$$

$\vert{\bf S}^{m-2}\vert$ is the measure of the unit Euclidean sphere ${\bf S}^{m-2},$ $\lambda = (m-2)/2,\ \ \tau_{n,\lambda }\$ is the first positive zero of the Gegenbauer's cosine polynomial $C^{\lambda }_n (\cos t)$.