Application of a Taylor series to approximate a function with large gradients

The method of approximating functions by polynomials based on Taylor series expansion is widely known. However, the residual term of such an approximation can be significant if the function has large gradients. The work assumes that the function has a decomposition in the form of a sum of regular and boundary layer components. The boundary layer component is a function of general form, known up to a factor, and is responsible for large gradients of the given function. This decomposition is valid, in particular, for the solution of a singularly perturbed problem. To approximate the function, a formula is proposed that uses the Taylor series expansion of the function and is exact for the boundary layer component. Under certain restrictions on the boundary layer component, estimates of the error in the approximation of the function are obtained. These estimates do not depend on the boundary layer component. Cases of functions of one and two variables are considered.