Approximation of Smooth Functions by Weighted Means of N-Point Padé 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 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.