S. R. Nasyrov’s Problem of Approximation by Simple Partial Fractions on an Interval

Abstract

In 2014, S. R. Nasyrov asked whether it is true that simple partial fractions (logarithmic derivatives of complex polynomials) with poles on the unit circle are dense in the complex space \(L_2[-1,1]\). In 2019, M. A. Komarov answered this question in the negative. The present paper contains a simple solution of Nasyrov’s problem different from Komarov’s one. Results related to the following generalizing questions are obtained: (a) of the density of simple partial fractions with poles on the unit circle in weighted Lebesgue spaces on \([-1,1]\); (b) of the density in \(L_2[-1,1]\) of simple partial fractions with poles on the boundary of a given domain for which \([-1,1]\) is an inner chord.