Weighted Lagrange interpolationwith pressigned nodes on the real line

In a recent paper, we investigated the uniform convergence of Lagrange interpolation at the zeros of the orthogonal polynomials with respect to a Freud-type weight in the presence of constraints. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial with respect to the given constraints well approximates a given function. Here, starting from the interest to construct a suitable interpolation operator with a preassigned node, we introduce an algorithm that allows us to obtain new matrices. For the Lagrange operator related to these new matrices that have the preassigned node among their elements, we prove results about the optimal rate of convergence as well as we apply successfully this method to some applications.