Approximation by Simplest Fractions

Abstract

In this paper, a number of problems concerning the uniform approximation of complex-valued continuous functions \(f\left( z \right)\) on compact subsets of the complex plane by simplest fractions of the form \(\Theta _n \left( z \right) = \sum\nolimits_{j = 1}^n {{1 \mathord{\left/ {\vphantom {1 {\left( {z - z_j } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {z - z_j } \right)}}}\) are considered. In particular, it is shown that the best approximation of a function \(f\) by the fractions \(\Theta _n\) is of the same order of vanishing as the best approximations by polynomials of degree \( \leqslant n\).