This document presents research on best one-sided approximation of multivariate functions in weighted spaces using multivariate Lagrange interpolation. Key points include: - It defines weighted spaces and best one-sided approximation for functions in these spaces. The degree of best one-sided approximation is minimized when the domain is a rectangle. - It constructs an operator for Lagrange interpolation and proves properties about best one-sided approximation using this operator. - The main results show that the degree of best one-sided approximation of a function's derivative is equal to the degree of best approximation of the function when restricted to a rectangle domain. - It also establishes an inverse theorem for best one-sided approximation using properties of best approximation and