Approximation of Smooth Functions by Weighted Means of N-Point Pad, Approximants

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 Pad, 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 alpha from the equation s = alpha s (1) + (1 - alpha)s (2) . Applying this weight to calculate the weighted mean alpha f (1) + (1 - alpha)f (2) we obtain significantly improved approximation of f.