An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces

Let $L^2_{\alpha,\beta}$ be the Hilbert space of real-valued functions on $[0,\pi]$ with scalar product
$$ (F,G)=\int_{0}^{\pi}F(x)G(x)\biggl(\sin\dfrac{x}{2}\biggr)^{2\alpha+1} \biggl(\cos\dfrac{x}{2}\biggr)^{2\beta+1}\,dx,\qquad \alpha>-1,\quad \beta>-1, $$
and norm $\|F\|=(F,F)^{1/2}$. We prove in the case when $\alpha>\beta\geqslant-1/2$ the following exact Jackson–Stechkin inequality
$$ E_{n-1} (F)\leqslant\omega_r\bigl(F,2x_{n}^{\alpha,\beta}\bigr),\quad F\in L^2_{\alpha,\beta}, $$
between the best of $F$ by cosine-polynomials of order $n-1$ and its generalized modulus of continuity of (real) order $r\geqslant 1$: $n\geqslant\max\bigl\{2,1+ \frac{\alpha-\beta}{2}\bigr\}$ if $\beta>-\frac12$ , $n\geqslant 1$ if $\beta=-\frac12$ , where $x_{n}^{\alpha,\beta}$ is the first positive zero of the Jacobi cosine-polynomial $P_{n}^{(\alpha,\beta)}(\cos x)$. We deduce from this inequality similar inequalities for mean-square approximations of functions of several variables given on projective spaces.