On the Continuity of the Sharp Constant in the Jackson–Stechkin Inequality in the Space $L^2$

This paper deals with the continuity of the sharp constant $K(T,X)$ with respect to the set $T$ in the Jackson–Stechkin inequality
$$ E(f,L)\le K(T,X)\omega(f,T,X), $$
where $E(f,L)$ is the best approximation of the function $f\in X$ by elements of the subspace $L\subset X$, and $\omega$ is a modulus of continuity, in the case where the space $L^2(\mathbb T^d,\mathbb C)$ is taken for $X$ and the subspace of functions $g\in L^2(\mathbb T^d,\mathbb C)$, for $L$. In particular, it is proved that the sharp constant in the Jackson–Stechkin inequality is continuous in the case where $L$ is the space of trigonometric polynomials of $n$th order and the modulus of continuity $\omega$ is the classical modulus of continuity of $r$th order.