%0 Journal Article
%T A New Extended Pad¨¦ Approximation and Its Application
%A Z. Kalateh Bojdi
%A S. Ahmadi-Asl
%A A. Aminataei
%J Advances in Numerical Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/263467
%X We extend ordinary Pad¨¦ approximation, which is based on a set of standard polynomials as , to a new extended Pad¨¦ approximation (M¨¹ntz Pad¨¦ approximation), based on the general basic function set £¿£¿ (the particular case of M¨¹ntz polynomials) using general Taylor series (based on fractional calculus) with error convergency. The importance of this extension is that the ordinary Pad¨¦ approximation is a particular case of our extended Pad¨¦ approximation. Also the parameterization ( is the corresponding parameter) of new extended Pad¨¦ approximation is an important subject which, obtaining the optimal value of this parameter, can be a good question for a new research. 1. Introduction Rational approximations of an arbitrary function are an important topic in numerical analysis due to their high applications in physical sciences, chemistry, engineering, and other applied sciences [1, 2]. The Pad¨¦ approximation is a particular and classical type of rational fraction approximation. The idea of this approximation is to expand a function as a ratio of two power series and determining both the numerator and denominator coefficients using the coefficients of Taylor series expansion of a function [1]. The Pad¨¦ approximation is the best approximation of a function by a rational function of a given order [1]. The technique was developed around 1890 by Henri Pad¨¦, but it goes back to George Freobenius who introduced the idea and investigated the features of rational approximations of power series. The Pad¨¦ approximation is usually superior when functions contain poles, because the use of rational function allows them to be well represented [1]. The Pad¨¦ approximation often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge [1]. For these reasons, Pad¨¦ approximation is used extensively in computer calculations. The Pad¨¦ approximation has also been used as an auxiliary function in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods in some sense inspired by the Pad¨¦ theory typically replace them. Since it provides an approximation to the function throughout the whole complex plane, the study of Pad¨¦ approximation is simultaneously a topic in mathematical approximation theory and analytic function theory. The generalized Pad¨¦ approximation is given in [2]. For the connection of Pad¨¦ approximation with continued fractions and orthogonal polynomials, see [2]. Also multivariate Pad¨¦ approximation was done by [3]. Two versions of this
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