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.