Nonuniqueness of best approximating complex rational functions

Approximating poles of complex rational functions

In this paper we investigate the application of the Nelder– Mead simplex method to approximate poles of complex rational functions. To our knowledge, there isn’t any algorithm which is able to find the poles of a function when only the values on the unit circle are given. We will show that this method can accurately approximate 1, 2 or even 3 poles without any preliminary knowledge of their loc...

The 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.

Series Representations for Best Approximating Entire Functions of Exponential Type

Let > 0 not be an even integer. We derive Lagrange type series representations for the entire function of exponential type 1 that minimizes jjjxj f (x)jjLp(R) amongst all such entire functions f , when p = 1 and p =1. This minimum arises as the scaled limit of the Lp error of polynomial approximation of jxj on [ 1; 1], and is one representation of the Lp Bernstein constant.

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class Rn,m of all rational functions of order (n,m) are considered. Let ρn,m = ρn,m( f ; E) be the distance of f in the uniform metric on E from the class Rn,m . We obtain results characterizing the rate of convergence to zero of the sequen...

The Existence of Rational Functions of Best Approximation*