Approximation of smooth functions by weighted means of n-point padé approximants

Abstract

Let f be a function we wish to approximate on the interval [x 1,x N ] knowing p 1 1,p 2,.. ,p N coefficients of expansion of f at the points x 1,x 2,.. ,x N. We start by computing two neighboring N -point Padé approximants (NPAs) of f, namely f 1 = [m/n] and f 2 = [m − 1/n] of f. The second NPA is computed with the reduced amount of information by removing the last coefficient from the expansion of f at x 1. We assume that f is sufficiently smooth, (e.g. convex-like function), and (this is essential) that f 1 and f 2 bound f in each interval]x i,x i+1 [on the opposite sides (we call the existence of such two-sided approximants the two-sided estimates property of f). Whether this is the case for a given function f is not necessarily known a priori, however, as illustrated by examples below it holds for many functions of practical interest. In this case, further steps become relatively simple. We select a known function s having the two-sided estimates property with values s(x i) as close as possible to the values f(x i). We than compute the approximants s 1 = [m/n] and s 2 = [m − 1/n] using the values at points x i and determine for all x the weight function α from the equation s = αs 1 + (1 − α)s 2. Applying this weight to calculate the weighted mean αf 1 + (1 − α)f 2 we obtain significantly improved approximation of f.