Some properties of Wright Operators

TL;DR

This paper introduces a novel family of Wright-based positive linear operators $W_n^{(\\beta)}$ that generalize classical approximation operators via the Wright function. It derives explicit moment formulas up to the fourth order, establishes mapping properties on the function space $E$, and provides quantitative rates of convergence via the modulus of continuity. The work further develops $A$-statistical convergence theory for these operators, proving a Korovkin-type theorem and a Voronovskaya-type asymptotic formula in the $A$-statistical sense. These results extend operator approximation theory with Wright-function kernels, offering robust convergence guarantees in both classical and statistical frameworks.

Abstract

This note aims to present novel positive linear operators involving the Wright function. Furthermore, the present research established the moments of these newly defined operators and estimated the convergence rate using the classical modulus of continuity. Additionally, the convergence rate in the Lipschitz spaces, their A-statistical convergence property, and $(λ, γ)$-statistical convergence properties have been covered.

Theorems & Definitions (16)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• proof
• Theorem 2