Behavior of Polynomials of Best Uniform Approximation
نویسندگان
چکیده
We investigate the asymptotic behavior of the polynomials {Pn(f)}'t' of best uniform approximation to a function f that is continuous on a compact set K of the complex plane C and analytic in the interior of K, where K has connected complement. For example, we show that for "most" functions f, the error f -Pn(f) does not decrease faster at interior points of K than on K itself. We also describe the possible limit functions for the normalized error (f -Pn(f))/En , where En := III -Pn(f)IIK, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to f on K exist that converge more rapidly at the interior points of K .
منابع مشابه
A method to obtain the best uniform polynomial approximation for the family of rational function
In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...
متن کاملThe best uniform polynomial approximation of two classes of rational functions
In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the results.
متن کاملPolynomials of the best uniform approximation to sgn(x) on two intervals
We describe polynomials of the best uniform approximation to sgn(x) on the union of two intervals [−A,−1] ∪ [1, B] in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error of this approximation. MSC 41A10, 41A25, 30C20.
متن کاملSolving Differential Equations by Using a Combination of the First Kind Chebyshev Polynomials and Adomian Decomposition Method
In this paper, we are going to solve a class of ordinary differential equations that its source term are rational functions. We obtain the best approximation of source term by Chebyshev polynomials of the first kind, then we solve the ordinary differential equations by using the Adomian decomposition method
متن کاملThe best approximation of some rational functions in uniform norm
Here we are concerned with the best approximation by polynomials to rational functions in the uniform norm. We give some new theorems about the best approximation of 1/(1 + x) and 1/(x − a) where a > 1. Finally we extend this problem to that of computing the best approximation of the Chebyshev expansion in uniform norm and give some results and conjectures about this. 2005 IMACS. Published by...
متن کامل