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Ukrainian Mathematical Journal

, Volume 65, Issue 10, pp 1566–1576 | Cite as

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

  • R. Jedynak
  • J. Gilewicz
Brief Communications

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.

Keywords

Weight Function Smooth Function Reference Function Weighted Approximation Hermite Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • R. Jedynak
    • 1
  • J. Gilewicz
    • 2
  1. 1.University Technolog.-Humanist.RadomPoland
  2. 2.Aix-Marseille University, CNRS, CPT, UMR 7332MarseilleFrance

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