Explicit solution of the finite time L2-norm polynomial approximation problem

Dedicated to Professor Alfredo Eisinberg on the Occasion of his 70th birthday Keywords: Polynomial approximation L 2-norm Hilbert matrix Laguerre polynomials Bernstein polynomials a b s t r a c t The aim of this paper is to present a new approach to the finite time L 2-norm polynomial approximation problem. A new formulation of this problem leads to an equivalent linear system whose solution can be investigated analytically. Such a solution is then specialized for a polynomial expressed in terms of Laguerre and Bernstein basis. In many applications fields it is very important to model signals of interest by polynomials (see [1,2] and the references therein). This research field is widely investigated and a large amount of efficient approaches have been proposed. The problem to find a good polynomial approximation is relevant in mathematical functions evaluation when software or hardware implementation are required [3]. Further applications can be found in speech compression [4], raw image encoding [5], control theory [6,7], to name just a few. Roughly speaking, the first natural target for every polynomial approximation scheme is to obtain a polynomial which is sufficiently close to a given function. Obviously, the property of closeness depends on the a priori chosen criteria to measure the quality of the approximation. In this paper, a different approach is presented to solve the polynomial approximation problem in finite time L 2-norm. Statistical considerations show that L 2-norm is the most appropriate choice for data fitting when the errors in the data have a normal distribution [1]. The practical relevance of considering such a norm is pointed out in several papers [8–10]. Let y(t) be the function to be approximated, then the L 2-norm polynomial approximation problem is formulated as follows.