Extremal and approximation properties of simple partial fractions

TL;DR

This paper surveys extremal and approximation properties of simple partial fractions (SPF), discussing classical and recent problems, methods, and results in approximation theory related to SPF and their derivatives.

Contribution

It systematically organizes known problems, results, and methods concerning SPF, providing a comprehensive overview and outlining key approaches in the field.

Findings

Systematization of extremal problems for SPF

Outline of methods for approximation and interpolation by SPF

Summary of inequalities and derivative estimation results

Abstract

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPF) as suggested by Eu.P. Dolzhenko. Many solved and unsolved extremal problems related to SPF are traced back to works of G. Boole, A.J. Macintyre, W.H.J. Fuchs, J.M. Marstrand, E.A. Gorin, A.A. Gonchar, Eu.P. Dolzhenko. At present, many authors systematically develop methods for approximation and interpolation by SPF and several their modifications. Simultaneously, related problems, being of independent interest, arise for SPF: inequalities of different metrics, estimation of derivatives, separation of singularities, etc. We systematize some of these problems which are known to us in Introduction of this survey. In the main part, we formulate principal results and outline methods to prove them if possible.