Polynomials Of Best Uniform Approximation Research Articles

On the Convergence of Best Uniform Deviations

If a function f is continuous on a closed Jordan curve $\Gamma$ and meromorphic inside $\Gamma$, then the polynomials of best uniform approximation to f on $\Gamma$ converge interior to $\Gamma$. Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of $\Gamma$. Applications and generalizations of this result are also given.

On the convergence of best uniform deviations

If a function f is continuous on a closed Jordan curve Γ \Gamma and meromorphic inside Γ \Gamma , then the polynomials of best uniform approximation to f on Γ \Gamma converge interior to Γ \Gamma . Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of Γ \Gamma . Applications and generalizations of this result are also given.

A characterization of badly approximable functions

Complex valued functions, continuous on a closed Jordan curve in the plane and having the property that all of their polynomials of best uniform approximation on that curve are identically zero are characterized in terms of their mapping properties on that curve.

Uniqueness of best approximation by monotone polynomials

The purpose of this paper is twofold. First, we prove the uniqueness of a polynomial of best uniform approximation, in a certain class P of “monotone” polynomials, to a given continuous function. This is the content of Theorem 3.1 which complements the results of Lorentz and Zeller [l]. Secondly, we prove (Theorem 6.1) that a polynomial of best L, approximation in the class 8, to a given continuous function is also unique. This is the analog of Jackson’s theorem for general polynomials. As a preliminary to Theorem 6.1, we give a necessary condition for a polynomial in B to be a polynomial of best L, approximation to an integrable function.

Estimates for Best Approximation to Rational Functions

Estimates for the deviation of certain rational functions and their polynomials of best uniform approximation on various sets are given. As a result, in some cases these deviation and polynomials are explicitly calculated. For example, the polynomials of best uniform approximation to the function $(\alpha z + \beta )/(z - a)(1 - \bar az),|a| \ne 1$, on the unit circle are given.

Estimates for best approximation to rational functions

Estimates for the deviation of certain rational functions and their polynomials of best uniform approximation on various sets are given. As a result, in some cases these deviation and polynomials are explicitly calculated. For example, the polynomials of best uniform approximation to the function ( α z + β ) / ( z − a ) ( 1 − a ¯ z ) , | a | ≠ 1 (\alpha z + \beta )/(z - a)(1 - \bar az),|a| \ne 1 , on the unit circle are given.

Overdetermined Systems of Linear Equations

for all x we call a Cebysev (approximate) solution (for short, C.S.) of (1).3 This report is a study of the properties of Cebysev solutions of (1). Most of the results we shall obtain may be found, explicitly or implicitly, in the literature. In particular this exposition relies on the presentations of Rivlin and Shapiro [1] and of Rademacher and Schoenberg [51, where more general results are obtained with the use of somewhat heavier machinery. The results we shall obtain go back to de La Vall6e Poussin [2]. We develop these results using only convexity ideas and some simple algebraic concepts. It is hoped that this approach helps unify the material and reaches the heart of the matter rather quickly. These same results are also to be found in a geometric and constructive setting in the excellent paper of Stiefel [8]. This is the appropriate place to mention the kinds of problems that lead to overdetermined systems of linear equations. Let f(t) be a given continuous realvalued function on a < t < b, and let 'k be the set of all polynomials (with real coefficients) of degree k. A central aspect of approximation theory is the study of the polynomial of best uniform approximation to f out of 'k . In recent years,

Polynomials of best uniform approximation to certain rational functions

Polynomials of best uniform approximation to certain rational functions