Bernstein-Jackson-type inequalities with exact constants in Orlicz spaces - M.I. Dmytryshyn; L.I. Dmytryshyn (Karpatsʹkì Matematičnì Publìkacìï, 2022)
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 $\mathcal{B}_τ^s(M)$ 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 $N_{ϑ,q}$ that is determined by the parameters $τ$ and $s$ of the approximation space $\mathcal{B}_τ^s(M)$.