The Jackson–Stechkin inequality with nonclassical modulus of continuity

We obtain an estimate for the best mean-square approximation $E_{n-1}(f)$ of an arbitrary complex-valued $2\pi$-periodic function $f\in L_{2}$ by the subspace $\Im_{2n-1}$ of trigonometric polynomials of degree at most $n-1$ in terms of the nonclassical modulus of continuity $\omega_{2m-1}^{*}(f,\delta)_{2}$ generated by a finite-difference operator of order $2m-1$ with alternating constant coefficients equal to 1 in absolute value. The following relation is proved for any natural $n\ge1$ and $m\ge2$:
$$ \sup_{\substack{f\in L_{2}\\ f\ne const}}\frac{E_{n-1}(f)}{\left(\displaystyle\frac{n}{2}\int_{0}^{\pi/n}\Big\{\omega_{2m-1}^{*}(f,t)\Big\}^{2}\sin ntdt\right)^{1/2}}={\frac{1}{\sqrt{2}}\Big(m-\sum\limits_{l=1}^{m-1}\frac{l}{4(m-l)^{2}-1}\Big)^{-1/2}}. $$