Best Approximation of Unbounded Functions by Modulus of Smoothness

In this paper, we study the approximation of unbounded functions in a weighted space by modulus smoothness using various linear operators. We establish direct theorems for such approximations and analyze properties within same space. Specifically, investigate behavior under different types operators, including Bernstein-Durrmeyer operator, Fejer Jackson operator. also provide detailed analysis convergence rate these Furthermore, discuss relationship between Lipschitz constant function. Our findings have important implications field theory may help to inform future research area.