Extension of the Best Polynomial Operator in Generalized Orlicz spaces

TL;DR

The paper develops an extension of the best polynomial $\varphi$-approximation operator from the Orlicz space $L^{\varphi}(\Omega)$ to the larger space $L^{\psi^{+}}(\Omega)$ for non-differentiable $\varphi$, by leveraging left/right derivatives $\psi^{-}, \psi^{+}$ and the Delta$_2$ condition. It first provides existence and a full characterization of the operator in $L^{\varphi}(\Omega)$ and then constructs the extended operator $\mu_{\psi^{+}}$ for $f \in L^{\psi^{+}}(\Omega)$, proving translation invariance and, under suitable monotonicity, uniqueness. A Calderón–Zygmund-type local smoothness class $t_m^{\psi^{+}}(x)$ is introduced to study pointwise convergence, yielding $P_x^{\varepsilon}(f) \to P_x(f)$ as $\varepsilon \to 0$ for functions with controlled local behavior. These results generalize prior differentiability-reliant treatments and connect generalized Orlicz approximation to Calderón–Zygmund theory, broadening applicability to non-smooth $N$-functions.

Abstract

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

Theorems & Definitions (32)

• Definition 1
• Theorem 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Theorem 2.4
• proof
• Remark 2.5
• Definition 2
• Definition 3