Published in: Izv. Ross. Akad. Nauk, Ser. Mat., 68 (1998), no. 6, pp. 27-52.;
English translation: Russian Acad. Sci. Math. Izv., 68 (1998), no. 6.

A B S T R A C T

B a b e n k o A. G.
Exact Jackson - Stechkin Inequality for $L^2$ - Approximations on the Segment with Jacobi Weight and on Projective Spaces

Let $L^2_{\alpha,\beta}$ be a Hilbert space of real functions on the segment $[0,\pi]$ with the inner product

$$(F,G)=\int_{0}^{\pi}F(x)G(x) \left(\sin \displaystyle{\frac{x... ...rac{x}{2}}\right)^{2\beta +1}dx, \quad \alpha>-1,\quad \beta>-1$$

$\mbox{and the norm}\ \Vert F\Vert=(F,F)^{1/2}.$ In the paper had been proved that exact Jackson - Stechkin inequality

$$E_{n-1} (F) \le \omega_r (F,2x_{n}^{\alpha,\beta}), \ F\in L^2_{\alpha,\beta},$$

$$n \ge \max\left\{2,1+\displaystyle{\frac{\alpha-\beta}{2}}\right\}\ \mbox{for}\ \beta>-1/2, \ n\ge 1\ \mbox{for}\ \beta=-1/2,$$

among the best approximation of the function $F$ by cosine polynomials of the order $n-1$ and its generalized modulus of continuity of the (real) order $r\ge 1$ is valid in the case $\alpha>\beta \ge -1/2.$ Here $x_{n}^{\alpha,\beta}$ is the first positive zero of Jacobi cosine polynomial $P_{n}^{(\alpha,\beta)}(\cos x).$

Analogouse results for mean squared approximations of functions of many variables given on the projective spaces are deduced from here.

Bibliogr. 49 titles.