Extension of the best polynomial approximation operator in Musielak—Orlicz spaces

In this paper, we considered the best polynomial approximation operator defined in the so-called Musielak—Orlicz space. Namely, we investigated the existence and uniqueness of the best polynomial approximation in Musielak—Orlicz space $L^{\Phi}$. We extended the best polynomial approximation operator from $L^{\Phi}$ to $L^{\varphi}$, where $\varphi(x,t)=\Phi'_{t+}(x,t)$. In addition, we established some properties of the extended best polynomial approximation operator.