# Approximation of Smooth Functions by Weighted Means of <em class="a-plus-plus">N</em>-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.*

Published

25.10.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 10, Oct. 2013, pp. 1410–1419, http://umj.imath.kiev.ua/index.php/umj/article/view/2518.

Issue

Section

Short communications