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

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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## Preview

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## References

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