Lagrange Interpolation Polynomial in Special Regions A Review

Abstract

It is known that Lagrange interpolation polynomial is considered as one of important tools to approximate the chosen function by a polynomial of degree  with knowing points that belong in it. Also, amount of error between the original function and this polynomial is known in books of approximation and numerical analysis. But studying the effects and amount of error of this polynomial in convex and compact regions has not been studied before. In this article, we will study in some detail the behavior of the Lagrange interpolation polynomial in such regions in terms of the form and its characteristics and the extent of the difference between this polynomial and the function in question. We will notice that this polynomial will be turned into a convex function when applied to a function in a convex region, and the degree of error or the amount of difference between it and the original function gets smaller if we apply our polynomial on the desired function in a convex region. The same is true if the effect of this polynomial is applied on a function in a compact region, and notice that the amount of difference between it and this function is reduced to zero if n goes to infinity.