An estimate of an optimal argument in the sharp multidimensional Jackson–Stechkin $L_2$-inequality

An estimate of an optimal argument in the sharp Jackson–Stechkin inequality in the space $L_2(\mathbb R^n)$ is proved in the case of a generalized modulus of continuity; its special case is the classical modulus of continuity. Similar statements hold for the torus $\mathbb T^n$. The obtained results agree with Chernykh's classical one-dimensional theorems and refine some results by S. N. Vasil'ev, A. I. Kozko, and N. I. Rozhdestvenskii.