A regularization method for polynomial approximation of functions from their approximate values at nodes

Abstract

A regularization method is proposed for the polynomial approximation of a function from its approximated values in a fixed family of nodes. As a regularization parameter we consider the number of nodes. We present explicit expressions for the optimal number of nodes in terms of the original error of the approximated values of the function. These problems appear frequently in studying inverse problems and when a smoothing technique should be applied to a series of numerical data. We obtain estimation of the approximation error by means of discrete versions of a convolution operators with polynomial kernels, and we observe the differences between the use of positive and non positive kernels.

Some numerical examples are provided to illustrate the efficiency and computational performance of the method. They also help us to compare different criteria for the construction of polynomial approximations.