Extension of the Best Polynomial Operator in Generalized Orlicz spaces

In this paper, we consider the best multivalued polynomial approximation operator for functions in an Orlicz Space $L^{\phi }(\mathrm{\Omega })$. We obtain its characterization involving ${\psi }^{-}$ and ${\psi }^{+}$, which are the left and right derivatives functions of $\phi$. And then, we extend the operator to $L^{{\psi }^{+}}(\mathrm{\Omega })$. We also get pointwise convergence of this extension, where the Calderón-Zygmund class $t_m^p(x)$ adapted to $L^{{\psi }^{+}}(\mathrm{\Omega })$ plays an important role.