A Remark on Chebyshev Polynomials Least Deviating from a Given Function

Abstract

In 1918 A. Haar [1] managed to state extremely general uniqueness conditions for a polynomial of best approximation. However, for thirty years since its publication, Haar’s result has not given rise to any new specific studies of analytical character. This seems to be due to the fact that no sufficiently simple and analytically natural cases of applicability of Haar’s conditions have been found outside the framework of “Chebyshev systems” in the sense of S.N. Bernshtein (see [2], §§1, 2 in). The work by Haar, along with the whole classical theory of best approximation, deals with real functions. In this paper we show that Haar’s theorem can be readily extended to complex functions. This makes it possible to apply it, for instance, to the problem of best approximation of a continuous complex function on any closed bounded set in the complex plane by ordinary polynomials (see [4]).