Uniform approximation by polynomials on compacta of special form
Let \(K\) be a compact subset of the complex plane, and \(f\) a continuous, complex-valued function defined on \(K\). This paper presents lower bounds for the best uniform approximation (on \(K\)) to \(f\) by a polynomial of a specified finite degree. The estimates depend on variations of concepts such as: the existence of polynomials \(P\) and \(Q\), complex numbers \(v_j \in K\) such that \(|(f \, P)(Q(v_j))| = \|f\, P\|\) and constructing measures on \(\{v_j\}\) that annihilate the linear space of approximating polynomials. The results are compared to, and improve, known results in the literature, and they are applied to established systems such as Chebyshev systems. However, the presentation in this translation obfuscates the results and examples.
scientific article; zbMATH DE number 6190790