Extension of the best polynomial operator in generalized Orlicz Spaces

Abstract

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