Zadorin A.I. Approximation of a function by polynomials in the presence of a region of large gradients

Abstract

The issue of approximating functions of one and two variables by polynomials in the presence of a region of large gradients is considered. The problem is that when applying Taylor's formula, the residual term can be significant if the function has large gradients. It is assumed that the function has a decomposition in the form of a sum of regular and boundary layer components. The boundary layer components are known up to a factor. This decomposition is valid for the solution of a singularly perturbed boundary value problem. The derivatives of the regular component are bounded to a certain order, and the boundary layer components have large gradients. Formulas for approximating a function by polynomials of an arbitrarily specified degree are constructed based on the fact that these formulas are exact for the boundary layer components. This approach has not been previously explored. Error estimates that are uniform in the boundary layer components are obtained.