Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces

We establish the Bernstein and Jackson type inequalities with exact constants for estimations of best approximations by exponential type functions in Orlicz spaces $L_M(\mathbb{R}^n)$. For this purpose, we use a special scale of approximation spaces $\mathcal{B}_\tau^s(M)$ that are interpolation spaces between the subspace $\mathscr{E}_M$ of exponential type functions and the space $L_M(\mathbb{R}^n)$. These approximation spaces are defined using a functional $E\left(t,f\right)$ that plays a similar role as the module of smoothness. The constants in obtained inequalities are expressed using a normalization factor $N_{\vartheta,q}$ that is determined by the parameters $\tau$ and $s$ of the approximation space $\mathcal{B}_\tau^s(M)$.